text:
\begin{document}

\title[Addendum to ``Extending invariant complex structures'']
{Addendum to: ``Extending invariant complex structures''}

\author[R. Campoamor Stursberg]{Rutwig Campoamor Stursberg}
\address{R. Campoamor Stursberg: I.M.I and Depto. Geometría y Topología, Universidad Complutense de Madrid, Spain
}
\email{rutwig@ucm.es}

\author[I. E. Cardoso]{Isolda E. Cardoso}
\address{I. E. Cardoso, ECEN-FCEIA, Universidad Nacional de Rosario \\
Pellegrini 250, 2000 Rosario, Santa Fe, Argentina}
\email{isolda@fceia.unr.edu.ar}

\author[G. P. Ovando]{Gabriela P. Ovando}
\address{G. P. Ovando: CONICET and ECEN-FCEIA, Universidad Nacional de Rosario \\
Pellegrini 250, 2000 Rosario, Santa Fe, Argentina}
\email{gabriela@fceia.unr.edu.ar}

\maketitle

In this note we write the explicit computations given in the work ``Extending complex structures'' (by R. Campoamor Stursberg, I. E. Cardoso and G. P. Ovando) in the Section Examples of totally real complex structures on six dimensional
Lie algebras.

The goal is to construct complex structures on six
dimensional Lie algebras $\ggo = \kk \oplus \hh$ where $\hh$ is an ideal of $\ggo$ isomorphic to $\RR^3$ and $\kk$ is a subalgebra of dimension three.

\medskip

 Recall first the classification of real  three dimensional Lie algebra $\kk=span\{e_1,e_2,e_3\}$, which are listed below  (see \cite{Mi} for instance):
$$
\begin{array}{ll}
\RR^3: & \mbox{ with trivial Lie bracket }\\
\hh_1: & [e_1,e_2]=e_3.\\
\rr_{3} & [e_1,e_2]=e_2,\quad [e_1,e_3]=e_2+e_3.\\
\rr_{3, \lambda}: & [e_1,e_2]=e_2,\quad [e_1,e_3]=\lambda e_3.\\
\rr_{3,\delta}: & [e_1,e_2]=e_2+\delta e_3\quad [e_1,e_3]=-\delta e_2+e_3.\\
\sso(3): & [e_1,e_2]=e_3,\quad [e_1,e_3]=-e_2, \qquad [e_2,e_3]=e_1.\\
\ssl(2): &  [e_1,e_2]=2, e_2\quad [e_1,e_3]=2 e_3,\quad [e_2,e_3]=e_1.
\end{array}
$$

\

{\bf  The construction.} Let $\kk$ be a three dimensional Lie algebra and let $\hh=\RR^3$
Let $\pi$ denote a representation from $\kk$ into $\hh$ by derivations. We search for $j\in Z^1(\kk,\pi)$ of maximal rank, that is $j:\kk \to \RR^3$ is an invertible linear operator satisfying  the linear equation
\begin{equation}\label{equ1}
0 =\pi(x) j(y)-\pi(y) j(x) - j([x,y]_{\kk})\qquad \mbox{ for all }x,y \in \kk.
\end{equation}

 This gives an integrable almost complex structure $J$ on $\ggo=\kk \ltimes \RR^3$, such thar both $\kk$ and $\hh$ as subspaces of $(\ggo, J)$ are totally real.

The complex structure $J$ on $\ggo$ is defined by $J_{|_{\kk}}=j$, $J_{|_{\hh}}=j^{-1}$. So if $\mathcal B$ is a basis of $\kk$ and $\mathcal B'$ is a basis of  $\hh$, then $\mathcal B \cup \mathcal B'$ is a basis of $\ggo$ with respect to which the complex structure $J$ has the form
 $$\left( \begin{matrix}0 & -j^{-1} \\
j & 0 \end{matrix} \right),
$$ where $j$ is a $ 3\times 3$ invertible real matrix. We shall write it with coefficients $j_{uv}$ as follows:
$$j= \left( \begin{matrix}
j_{41} & j_{42} & j_{43} \\
j_{51} & j_{52} & j_{53} \\
j_{61} & j_{62} & j_{63}
\end{matrix} \right)\qquad \quad \det j\neq 0.
$$

Next  we evaluate the equation (\ref{equ1}). If $\{e_1,e_2,e_3\}$ denotes a basis of $\kk$ and $\{e_4,e_5,e_6\}$ denotes a basis of $\hh$, Equation (\ref{equ1}) becomes

$$
\begin{array}{rcl}
0 & = & \pi(e_i) j(e_k) - \pi(e_k) j(e_i) - j [e_i,e_k]_{\kk}\\
& = &  \sum_{s=4}^6 j_{sk} \pi(e_i) e_s - \sum_{s=4}^6 j_{si} \pi(e_k) e_s
- j (\sum_{l=1}^3 C_{ik}^l) e_l
\end{array}
$$
which is a linear system on the coefficients $j_{uv}$, for a fixed representation by derivations $\pi$ of $\kk$ into $\RR^3$ and for $\{C_{ik}^l\}$ being the structure coefficients for $\kk$.  We shall choose a representation $\pi$ such  that $\Ima \pi=1$ to do explicit computations.

For $\hh =\RR^3$,  any representation $\pi:\kk \to \End(\RR^3)$  such that $\dim\Ima \pi=1$ is determined by a linear map $t$. Thus there is a basis of $\RR^3$ in which the matrix of $t$ is of one and only one of the following types

\begin{equation}\label{typet1} i)\, \left( \begin{matrix}
\eta & 0 & 0 \\
0 & \nu & 0 \\
0 & 0 & \mu
\end{matrix} \right)\qquad ii)\, \left( \begin{matrix}
\eta & 0 & 0 \\
0 & \nu & -\mu  \\
0 & \mu & \nu
\end{matrix} \right)\qquad iii)\,
\left( \begin{matrix}
\eta & 0 & 0 \\
0 & \nu & 1 \\
0 & 0 & \nu
\end{matrix} \right)\qquad iv)\,
\left( \begin{matrix}
\eta & 1 & 0 \\
0 & \eta & 1 \\
0 & 0 & \eta
\end{matrix} \right).
\end{equation}

\vspace{.3cm}

For $\hh =\hh_1$,  any representation $\pi:\kk \to \Der(\hh_1)$  such that $\dim\Ima \pi=1$ is determined by a derivation $t$     of $\hh_1$ . Thus there is a basis of $\hh_1$ in which the matrix of $t$ is of one and only one of the following types (see for instance \cite{ABDO}):

\begin{equation}\label{typet2} i)\, \left( \begin{matrix}
\eta & \nu & 0 \\
\mu & -\eta & 0 \\
0 & 0 & 0
\end{matrix} \right)\qquad \qquad ii)\, \left( \begin{matrix}
\eta & \nu & 0 \\
\mu & 1-\eta & 0  \\
0 & 0 & 1
\end{matrix} \right).
\end{equation}

In any case if  we assume that $\pi(e_1) = \varepsilon_1 t$, $\pi(e_2)=\varepsilon_2 t$, $\pi(e_3)=\varepsilon_3 t$, the condition of $\pi$ being a representation says
$$\phi([x,y]) =[\pi(x), \pi(y)]= 0$$
where the last equality holds due to $\Ima \pi=span\{t\}$.
Hence $\pi(x)=0$ for every $x\in C^1(\kk)$.

This explanation gives the proof of the following Lemma. See for instance \cite{Va} for representations of $\ssl(2)$.

\begin{lem} The simple Lie algebras $\ssl(2)$ and $\sso(3)$ do not admit any representation $\pi:\kk\to\End(V)$ such that $\dim \Ima \pi=1$.
\end{lem}

For the solvable Lie algebras $\hh_1, \rr_3, \rr_{3,\lambda}, \rr_{3,\delta}$ one has $\pi(e_j)=0$ for $e_j\in C^1(\kk)$. Since in all these cases $e_1\in\kk-C^1(\kk)$ we shall assume $\pi(e_1)=t$ and we take $\pi(e_2)=\varepsilon_1 t, \pi(e_3)=\varepsilon_2 t$, where $\varepsilon_i$ could be zero as explained above.

The computations for the proof of the next theorems can be done with help of a software.

\begin{thm} Let $\kk$ be a three dimensional solvable Lie algebra. Let $\pi:\kk \to \End(\RR^3)$ be a representation  such that $\Ima \pi=span\{t\}$ where $t$ is as in (\ref{typet1}). Then the semidirect product Lie algebra $\ggo=\kk\oplus_{\pi} \RR^3$ admits a  complex structure $J$ such that $J\kk=\RR^3$ in the cases exposed in  Table 1.
\end{thm}

\begin{thm} Let $\kk$ be a three dimensional solvable Lie algebra. Let $\pi:\kk \to \End(\hh_1)$ be a representation acting by derivations on the Heisenberg Lie algebra $\hh_1$  with $\dim \Ima \pi= 1$ as in (\ref{typet2}).

Then there exists a Lie algebra  with a  complex structure $(\ggo, J)$ such that $\hh_1$ is a totally real ideal of the   extended semidirect product Lie algebra $\ggo=\kk\oplus_{\pi} \hh_1$
(as in  Proposition 3.6 of ``Extending complex structures'') in the cases exposed in Table 2.

\end{thm}

\smallskip

{\bf  The results.} Next we present the results of these computations done as follows: for each Lie algebra $\hh=\RR^3$ and $\hh=\hh_1$, and for each Lie algebra $\mathfrak{k}$ (cases I to IV) we show the solutions, if there are solutions, for each type of representation (types i to iv). Each solution consists of a matrix with two horizontal entries: the above entry shows the form of the matrix $j$ and its determinant, which we consider nonzero, and the below entry shows the parameters $\eta,\nu,\mu,\varepsilon_{1},\varepsilon_{2},\lambda$ and $\delta$, wherever they correspond. Also, we remark that some of the solutions are complex, which we do not consider as a valid solution for our problem but we present them anyway for the sake of completeness. Finally, below the explicit solutions we present the table, depicted in the article, where we summarize this computations.

\subsection{Computations for $\hh=\RR^3$}

\begin{enumerate}

\item CASE I: $\mathfrak{k}=\mathfrak{h}_{1}$.
		\begin{enumerate}
			\item For a type (i) representation: there are no solutions.
			\item For a type (ii) representation: there are no solutions.
			\item For a type (iii) there are 2 solutions.
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{53}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{{\it j_{53}}}^{2}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\eta=0,\nu=0,\varepsilon=\varepsilon]\end {array}
 \right]
                              $$

                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{53}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{{\it j_{53}}}^{2}]\\ \noalign{\medskip}[\eta=\eta,\nu=0
,\varepsilon=\varepsilon]\end {array} \right]
                              $$

			\item For a type (iv) representation: there is 1 solution.
			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&\varepsilon\,{
\it j_{51}}+{\it j_{43}}&0\\ \noalign{\medskip}{\it j_{61}}&\varepsilon\,{\it j_{61}}
&0\end {array} \right] ,{\it det(j)}=-{{\it j_{43}}}^{2}{\it j_{61}}]
\\ \noalign{\medskip}[\eta=0,\varepsilon=\varepsilon]\end {array} \right]
                              $$

		\end{enumerate}

\item CASE II: $\mathfrak{k}=\mathfrak{r}_{3}$.
		\begin{enumerate}
			\item For a type (i) representation: there are no solutions.
			\item For a type (ii) representation: there are no solutions.
			\item For a type (iii) there are 2 solutions.
			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{52}}\end {array} \right] ,{\it
det(j)}={\it j_{41}}\,{{\it j_{52}}}^{2}-{\it j_{61}}\,{\it j_{43}}\,{\it j_{52}}]
\\ \noalign{\medskip}[\eta=1,\nu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{52}}\end {array} \right] ,{\it
det(j)}={\it j_{41}}\,{{\it j_{52}}}^{2}]\\ \noalign{\medskip}[\eta=\eta,\nu=1
]\end {array} \right]
                              $$
			\item For a type (iv) there is 1 solution.
			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{53}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}={{
\it j_{53}}}^{2}{\it j_{61}}]\\ \noalign{\medskip}[\eta=1]\end {array}
 \right]
                              $$

\end{enumerate}

\item CASE III: $\mathfrak{k}= \mathfrak{r}_{3,\lambda}$.
		\begin{enumerate}
			\item For a type (i) representation: there are 28 solutions.
		                                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&{
\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
\det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}
]\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=0,\mu=1,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&{
\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
\det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=1,\mu=0,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}=  \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}\\ \noalign{\medskip}{
\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it \det(j)}=-{\it j_{42}}\,{\it
j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[
\lambda=0,\eta=1,\nu=0,\mu=0,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&0
\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0\\ \noalign{\medskip}{\it
j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it \det(j)}={\it j_{41}}\,{\it j_{52}}
\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=1,\mu=0,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}=  \left[ \begin {array}{ccc} {\it j_{41}}&0&0
\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}\\ \noalign{\medskip}{\it
j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it \det(j)}=-{\it j_{41}}\,{\it j_{53}}
\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=0,\mu=1,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&0\\ \noalign{\medskip}{\it j_{61}}
&0&{\it j_{63}}\end {array} \right] ,{\it \det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{
\it j_{63}}]\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=\nu,\mu=0,\varepsilon
=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}\\ \noalign{\medskip}{
\it j_{61}}&0&0\end {array} \right] ,{\it \det(j)}={\it j_{42}}\,{\it j_{53}}\,{
\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=0,\mu=\mu,\varepsilon
=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&{
\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0\\ \noalign{\medskip}{\it
j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it \det(j)}={\it j_{43}}\,{\it j_{51}}
\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=\nu,\mu=1,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&{
\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it \det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=0,
\nu=1,\mu=\mu,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}\\ \noalign{\medskip}{
\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it \det(j)}=-{\it j_{41}}\,{\it
j_{53}}\,{\it j_{62}}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[
\lambda=0,\eta=1,\nu=0,\mu=1,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0\\ \noalign{\medskip}{
\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it \det(j)}={\it j_{41}}\,{\it
j_{52}}\,{\it j_{63}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}]\\ \noalign{\medskip}[
\lambda=0,\eta=1,\nu=1,\mu=0,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ {\it j}= \left[ \begin {array}{ccc} {\it j_{41}}&0&{
\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
\det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=1,\mu=1,\varepsilon=0]
\end {array} \right]
                              $$

			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda
\neq 0,\lambda=\mu,\eta=1,\nu=\nu,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}={
\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=\nu,\eta=1,\nu=\nu,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda
\neq 0,\lambda=\lambda,\eta=\lambda,\nu=\nu,\mu=1]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=\lambda,\eta=\lambda,\nu=1,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda
\neq 0,\lambda=\mu,\eta=\eta,\nu=1,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda
\neq 0,\lambda=\nu,\eta=\eta,\nu=\nu,\mu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\nu,\eta=1,\nu=\nu,\mu=
\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\mu,\eta=1,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\nu,\eta=1,\nu=\nu,\mu=1]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\mu,\eta=\mu,\nu=1,\mu=\mu
]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\nu,\eta=\nu,\nu=\nu,\mu=
1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=\lambda,\nu=1
,\mu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-{\it j_{51}}\, \left( {\it j_{42}}\,{\it j_{63}}-{\it j_{43}
}\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,
\eta=1,\nu=\nu,\mu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it
det(j)}={\it j_{61}}\, \left( {\it j_{42}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{52}}
 \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,\nu=1,
\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,
\eta=\eta,\nu=1,\mu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{41}}\,{\it
j_{53}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,
{\it j_{61}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it
j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,\nu=1,\mu=1]
\end {array} \right]
                              $$

			\item For a type (ii) representation: there are 26 solutions.
			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=1/2,\mu=1/2\,i,\varepsilon=0
]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=1/2,\mu=-1/2\,i,\varepsilon
=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}-{\it j_{51}}\,{
\it j_{42}}\,{\it j_{63}}+i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=1/2,\mu=1/2\,i,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}-{\it j_{51}}\,{
\it j_{42}}\,{\it j_{63}}-i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=1/2,\mu=-1/2\,i,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{51}}\,{\it j_{42}}\,{\it j_{63}}+{\it j_{61}}\,{\it j_{42}}\,{\it j_{53}}
]\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=0,\mu=0,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}+{\it j_{43}}\,{
\it j_{51}}\,{\it j_{62}}+i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=1/2,\mu=1/2\,i,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}+{\it j_{43}}\,{
\it j_{51}}\,{\it j_{62}}-i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=1/2,\mu=-1/2\,i,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=1,\mu=0,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{41}}\,{\it
j_{53}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,
{\it j_{61}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it
j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,\nu=1,\mu=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,
\eta=\eta,\nu=1,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=\eta,\nu=\nu
,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=\eta,\nu=\nu
,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&-i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}-i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}
}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=1,\nu=\nu,
\mu=i \left( \lambda-\nu \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&i{\it j_{63}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}
}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=1,\nu=\nu,
\mu=-i \left( \lambda-\nu \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\nu,\eta=1,\nu=\nu,\mu=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+i{\it j_{61}}\,{
\it j_{42}}\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta
=1,\nu=\nu,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}-i{\it j_{61}}\,{
\it j_{42}}\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta
=1,\nu=\nu,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}-{\it j_{42}}\,{
\it j_{51}}\,{\it j_{63}}-i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=1,\nu=\nu,
\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}-{\it j_{42}}\,{
\it j_{51}}\,{\it j_{63}}+i{\it j_{61}}\,{\it j_{42}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=1,\nu=\nu,
\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=\lambda,\nu=
\nu,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}+i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=\lambda,\nu=
\nu,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta=\lambda,\nu=1
,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-i{\it j_{61}}\,{\it
j_{43}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,
\nu=\nu,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}+i{\it j_{61}}\,{\it
j_{43}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,
\nu=\nu,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&-i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{62}}\,{\it j_{63}}+{\it j_{43}}\,{
\it j_{51}}\,{\it j_{62}}-i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=2\,\nu-1,\nu
=\nu,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}&i{\it
j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{63}}\,{\it j_{62}}+{\it j_{43}}\,{
\it j_{51}}\,{\it j_{62}}+i{\it j_{61}}\,{\it j_{43}}\,{\it j_{62}}]
\\ \noalign{\medskip}[\lambda\neq 0,\lambda=2\,\nu-1,\eta=2\,\nu-1,\nu
=\nu,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$

			\item For a type (iii) there are 5 solutions.
			                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}={
\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=1,
\nu=0,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=0,
\nu=1,\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}={
\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=\nu,\eta=1,\nu=\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=\lambda,\eta=\lambda,\nu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it
det(j)}={\it j_{61}}\, \left( {\it j_{42}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{52}}
 \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,\eta=1,\nu=1]
\end {array} \right]
                              $$

			\item For a type (iii) there are no solutions.
		\end{enumerate}

\item CASE IV: $\mathfrak{k}=\mathfrak{r}_{3,\delta}$.
		\begin{enumerate}
			\item For a type (i) representation: there are 16 solutions.

                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\delta=0,\eta=\eta,\nu=0,
\mu=1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{41}}\,{\it
j_{53}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,
{\it j_{61}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it
j_{61}}]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=0,\mu=1]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-{\it j_{51}}\, \left( {\it j_{42}}\,{\it j_{63}}-{\it j_{43}
}\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=0,\mu=
1]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it
det(j)}={\it j_{61}}\, \left( {\it j_{42}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{52}}
 \right) ]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=0,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{
\it det(j)}=-2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[
\delta=-i \left( \eta-1 \right) ,\eta=\eta,\nu=-i \left( \eta-1
 \right) ,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{
\it det(j)}=2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[
\delta=i \left( \eta-1 \right) ,\eta=\eta,\nu=i \left( \eta-1 \right)
,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=-i \left( \nu-1 \right) ,\eta=\eta,\nu=-i
 \left( \nu-1 \right) ,\mu=2-\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=i \left( \nu-1 \right) ,\eta=\eta,\nu=i
 \left( \nu-1 \right) ,\mu=2-\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=-i \left( \eta-1 \right) ,\eta=\eta,\nu=-
i \left( \eta-1 \right) ,\mu=-\eta+2]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=i \left( \eta-1 \right) ,\eta=\eta,\nu=i
 \left( \eta-1 \right) ,\mu=-\eta+2]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it
j_{53}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}+
2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]\\ \noalign{\medskip}[\delta=-i
 \left( \nu-1 \right) ,\eta=\nu,\nu=-i \left( \nu-1 \right) ,\mu=2-\nu
]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}}
&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}-2
\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]\\ \noalign{\medskip}[\delta=i
 \left( \nu-1 \right) ,\eta=\nu,\nu=i \left( \nu-1 \right) ,\mu=2-\nu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}-2
\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=i
 \left( \nu-1 \right) ,\eta=2-\nu,\nu=i \left( \nu-1 \right) ,\mu=\nu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}+
2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=-i
 \left( \nu-1 \right) ,\eta=2-\nu,\nu=-i \left( \nu-1 \right) ,\mu=2-
\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}-2
\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=i
 \left( \nu-1 \right) ,\eta=2-\nu,\nu=i \left( \nu-1 \right) ,\mu=2-
\nu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}+
2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=-i
 \left( \nu-1 \right) ,\eta=2-\nu,\nu=-i \left( \nu-1 \right) ,\mu=\nu
]\end {array} \right]
                              $$

			\item For a type (ii) representation: there are 16 solutions.

                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{63}}&
{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-{\it j_{53}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-i{\it j_{43}}\,{\it j_{51}}\,{\it j_{63}}+i{
\it j_{43}}\,{\it j_{53}}\,{\it j_{61}}+{\it j_{41}}\,{{\it j_{53}}}^{2}+{\it j_{41}}\,{
{\it j_{63}}}^{2}-{\it j_{43}}\,{\it j_{51}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{61}}\,{
\it j_{63}}]\\ \noalign{\medskip}[\delta=\mu,\eta=-i\mu+1,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{63}}
&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-{\it j_{53}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=i{\it j_{43}}\,{\it j_{51}}\,{\it j_{63}}-i{
\it j_{43}}\,{\it j_{53}}\,{\it j_{61}}+{\it j_{41}}\,{{\it j_{53}}}^{2}+{\it j_{41}}\,{
{\it j_{63}}}^{2}-{\it j_{43}}\,{\it j_{51}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{61}}\,{
\it j_{63}}]\\ \noalign{\medskip}[\delta=\mu,\eta=i\mu+1,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\it j_{63}}
&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{53}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-i{\it j_{43}}\,{\it j_{51}}\,{\it j_{63}}+i{
\it j_{43}}\,{\it j_{53}}\,{\it j_{61}}-{\it j_{41}}\,{{\it j_{53}}}^{2}-{\it j_{41}}\,{
{\it j_{63}}}^{2}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{53}}+{\it j_{43}}\,{\it j_{61}}\,{
\it j_{63}}]\\ \noalign{\medskip}[\delta=-\mu,\eta=i\mu+1,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\it j_{63}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{53}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=i{\it j_{43}}\,{\it j_{51}}\,{\it j_{63}}-i{
\it j_{43}}\,{\it j_{53}}\,{\it j_{61}}-{\it j_{41}}\,{{\it j_{53}}}^{2}-{\it j_{41}}\,{
{\it j_{63}}}^{2}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{53}}+{\it j_{43}}\,{\it j_{61}}\,{
\it j_{63}}]\\ \noalign{\medskip}[\delta=-\mu,\eta=-i\mu+1,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{62}}&
i{\it j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=i{\it j_{42}}\,{\it j_{61}}\,{\it j_{63}}-i{
\it j_{43}}\,{\it j_{61}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it
j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=
\nu,\mu=i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{62}}
&-i{\it j_{63}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-i{\it j_{42}}\,{\it j_{61}}\,{\it j_{63}}+i{
\it j_{43}}\,{\it j_{61}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it
j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=
\nu,\mu=-i \left( \nu-1 \right) ]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{63}}&
-{\frac {{\it j_{63}}\, \left( -i\nu+i \right) }{\nu-1}}
\\ \noalign{\medskip}{\it j_{61}}&{\frac {{\it j_{63}}\, \left( \mu\,
 \left( -i\nu+i+\mu \right) -{\mu}^{2}-{\nu}^{2}+2\,\nu-1 \right) }{
 \left( \nu-1 \right)  \left( -i\nu+i+\mu \right) }}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,{\frac {{\it j_{43}}\,{\it j_{63}}\,
 \left( i{\it j_{51}}\,\mu-i{\it j_{61}}\,\nu+i{\it j_{61}}+{\it j_{51}}\,\nu+{
\it j_{61}}\,\mu-{\it j_{51}} \right) }{i\nu-i-\mu}}]\\ \noalign{\medskip}[
\delta=-i\nu+i+\mu,\eta=-i \left( -i\nu+i+\mu \right) +1,\nu=\nu,\mu=
\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{63}}
&-{\frac {{\it j_{63}}\, \left( i\nu-i \right) }{\nu-1}}
\\ \noalign{\medskip}{\it j_{61}}&{\frac {{\it j_{63}}\, \left( \mu\,
 \left( i\nu-i+\mu \right) -{\mu}^{2}-{\nu}^{2}+2\,\nu-1 \right) }{
 \left( \nu-1 \right)  \left( i\nu-i+\mu \right) }}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,{\frac {{\it j_{43}}\,{\it j_{63}}\,
 \left( i{\it j_{51}}\,\mu-i{\it j_{61}}\,\nu+i{\it j_{61}}-{\it j_{51}}\,\nu-{
\it j_{61}}\,\mu+{\it j_{51}} \right) }{i\nu-i+\mu}}]\\ \noalign{\medskip}[
\delta=i\nu-i+\mu,\eta=i \left( i\nu-i+\mu \right) +1,\nu=\nu,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\it j_{63}}
&{\frac {{\it j_{63}}\, \left( -i\nu+i \right) }{\nu-1}}
\\ \noalign{\medskip}{\it j_{61}}&-{\frac {{\it j_{63}}\, \left(  \left( -i
\nu+i-\mu \right) \mu+{\mu}^{2}+{\nu}^{2}-2\,\nu+1 \right) }{ \left(
\nu-1 \right)  \left( -i\nu+i-\mu \right) }}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,{\frac {{\it j_{43}}\,{\it j_{63}}\, \left( i{\it
j_{51}}\,\mu-i{\it j_{61}}\,\nu+i{\it j_{61}}-{\it j_{51}}\,\nu-{\it j_{61}}\,\mu+{
\it j_{51}} \right) }{i\nu-i+\mu}}]\\ \noalign{\medskip}[\delta=-i\nu+i-
\mu,\eta=-i \left( -i\nu+i-\mu \right) +1,\nu=\nu,\mu=\mu]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\it j_{63}
}&{\frac {{\it j_{63}}\, \left( i\nu-i \right) }{\nu-1}}
\\ \noalign{\medskip}{\it j_{61}}&-{\frac {{\it j_{63}}\, \left(  \left( i
\nu-i-\mu \right) \mu+{\mu}^{2}+{\nu}^{2}-2\,\nu+1 \right) }{ \left(
\nu-1 \right)  \left( i\nu-i-\mu \right) }}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,{\frac {{\it j_{43}}\,{\it j_{63}}\, \left( i{\it
j_{51}}\,\mu-i{\it j_{61}}\,\nu+i{\it j_{61}}+{\it j_{51}}\,\nu+{\it j_{61}}\,\mu-{
\it j_{51}} \right) }{i\nu-i-\mu}}]\\ \noalign{\medskip}[\delta=i\nu-i-
\mu,\eta=i \left( i\nu-i-\mu \right) +1,\nu=\nu,\mu=\mu]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{63}}&-{\it j_{62}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {{\it j_{62}}}^{2}+{{\it j_{63}}}^{2
} \right) ]\\ \noalign{\medskip}[\delta=\mu,\eta=\eta,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&-{\it j_{63}}&{\it j_{62}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( -{{\it j_{62}}}^{2}-{{\it j_{63}}}^{
2} \right) ]\\ \noalign{\medskip}[\delta=-\mu,\eta=\eta,\nu=1,\mu=\mu]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\delta=0,\eta=\eta,\nu=1,
\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}+
2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=-i
 \left( \nu-1 \right) ,\eta=-\nu+2,\nu=\nu,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}-2
\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[\delta=i
 \left( \nu-1 \right) ,\eta=-\nu+2,\nu=\nu,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-{\it j_{41}}\,{\it
j_{53}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{\it j_{42}}\,{\it j_{53}}\,
{\it j_{61}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it
j_{61}}]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=1,\mu=0]\end {array}
 \right]
                              $$

			\item For a type (iii) representation: there are 3 solutions.

                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it
det(j)}={\it j_{61}}\, \left( {\it j_{42}}\,{\it j_{53}}-{\it j_{43}}\,{\it j_{52}}
 \right) ]\\ \noalign{\medskip}[\delta=0,\eta=1,\nu=1]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{
\it det(j)}=2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[
\delta=-i \left( \nu-1 \right) ,\eta=-\nu+2,\nu=\nu]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}
}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{
\it det(j)}=-2\,i{\it j_{61}}\,{\it j_{43}}\,{\it j_{53}}]\\ \noalign{\medskip}[
\delta=i \left( \nu-1 \right) ,\eta=-\nu+2,\nu=\nu]\end {array}
 \right]
                              $$

			\item For a type (iv) representation: there are no solutions.

\end{enumerate}

\end{enumerate}

 This gives the results in Table 1.

\medskip

\subsection{Computations for $\hh=\hh_1$}

\begin{enumerate}

\item CASE I: $\mathfrak{k}=\mathfrak{h}_{1}$.
		\begin{enumerate}
			\item For a type (i) representation: there are 2 solutions.
			                         $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&-\varepsilon\,\eta\,{\it j_{41}}-\varepsilon\,{\it j_{51}}\,\nu
+\eta\,{\it j_{42}}+\nu\,{\it j_{52}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}
}&{\frac {\eta\, \left( \varepsilon\,\eta\,{\it j_{41}}+\varepsilon\,{\it j_{51}}
\,\nu-\eta\,{\it j_{42}}-\nu\,{\it j_{52}} \right) }{\nu}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] , \\ \noalign{\medskip} {\it
\det(j)}=-{\frac { \left( \eta\,{\it j_{41}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{61}}
\,\eta+{\it j_{51}}\,{\it j_{62}}\,\nu-{\it j_{52}}\,{\it j_{61}}\,\nu \right)
 \left( \varepsilon\,\eta\,{\it j_{41}}+\varepsilon\,{\it j_{51}}\,\nu-\eta\,{\it
j_{42}}-\nu\,{\it j_{52}} \right) }{\nu}}]\\ \noalign{\medskip}[\eta=\eta,
\nu=\nu,\mu=-{\frac {{\eta}^{2}}{\nu}},\varepsilon=\varepsilon]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&-\varepsilon
\,{\it j_{41}}\,\mu+\mu\,{\it j_{42}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}
}&0\end {array} \right] ,{\it \det(j)}=\varepsilon\,{{\it j_{41}}}^{2}{\it j_{62}}
\,\mu-\varepsilon\,{\it j_{41}}\,{\it j_{42}}\,{\it j_{61}}\,\mu-{\it j_{41}}\,{\it
j_{42}}\,{\it j_{62}}\,\mu+{{\it j_{42}}}^{2}{\it j_{61}}\,\mu]
\\ \noalign{\medskip}[\eta=0,\nu=0,\mu=\mu,\varepsilon=\varepsilon]
\end {array} \right]
                              $$

			\item For a type (ii) representation: there are no solutions.
		\end{enumerate}

\item CASE II: $\mathfrak{k}=\mathfrak{r}_{3}$.
		\begin{enumerate}
			\item For a type (i) representation: there are no solutions.
			\item For a type (ii) representation: there are no solutions.
		\end{enumerate}

\item CASE III: $\mathfrak{k}= \mathfrak{r}_{3,\lambda}$.
		\begin{enumerate}
			\item For a type (i) representation: there are 6 solutions.
                              $$
 \left[ \begin {array}{c} [ \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&-{\frac {{\it j_{42}}\, \left( \eta-
1 \right) }{\nu}}&0\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-{\frac {{\it j_{42}}\,{\it j_{63}}\,
 \left( \eta\,{\it j_{41}}+{\it j_{51}}\,\nu-{\it j_{41}} \right) }{\nu}}]
\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=\nu,\mu=-{\frac {{\eta}^
{2}-1}{\nu}},\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ \left[ \begin {array}{ccc} {\it j_{41}}&{\it
j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&1/2\,{\it j_{42}}\,\mu&0
\\ \noalign{\medskip}{\it j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it
det(j)}=1/2\,{\it j_{41}}\,{\it j_{42}}\,\mu\,{\it j_{63}}-{\it j_{42}}\,{\it j_{51}}\,
{\it j_{63}}]\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=0,\mu=\mu,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [ \left[ \begin {array}{ccc} {\it j_{41}}&0&0
\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0\\ \noalign{\medskip}{\it
j_{61}}&0&{\it j_{63}}\end {array} \right] ,{\it det(j)}={\it j_{41}}\,{\it j_{52}}
\,{\it j_{63}}]\\ \noalign{\medskip}[\lambda=0,\eta=-1,\nu=0,\mu=\mu,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&-1/2\,{\it j_{53}}\,\nu\\ \noalign{\medskip}{\it j_{51}}&0
&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{
\it det(j)}={\it j_{42}}\,{\it j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[
\lambda\neq 0,\lambda=-1,\eta=1,\nu=\nu,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\frac {{\it j_{52}}\, \left( {\it j_{43}}\,\mu+2\,{\it j_{53}}
 \right) }{{\it j_{53}}\,\mu}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{
\it j_{52}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array}
 \right] ,{\it det(j)}=2\,{\frac {{\it j_{61}}\,{\it j_{52}}\,{\it j_{53}}}{\mu}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=-1,\eta={\frac {{\it j_{43}}
\,\mu+{\it j_{53}}}{{\it j_{53}}}},\nu=-{\frac { \left( {\it j_{43}}\,\mu+2\,{
\it j_{53}} \right) {\it j_{43}}}{{{\it j_{53}}}^{2}}},\mu=\mu]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&0
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=-1,\eta=-1,\nu=2\,{\frac {{\it j_{42}}}{{\it j_{52}}}},\mu=0]
\end {array} \right]
                              $$

			\item For a type (ii) representation: there are 11 solutions.
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&-{\frac {{\it j_{53}}\,\nu}{\eta}}
\\ \noalign{\medskip}{\it j_{51}}&-{\frac {{\it j_{42}}\, \left( \eta-1
 \right) }{\nu}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0
\end {array} \right] ,{\it det(j)}=-{\frac {{\it j_{53}}\, \left( \eta\,{
\it j_{41}}\,{\it j_{62}}+{\it j_{51}}\,{\it j_{62}}\,\nu-{\it j_{42}}\,{\it j_{61}}
 \right) }{\eta}}]\\ \noalign{\medskip}[\lambda=0,\eta=\eta,\nu=\nu,
\mu=-{\frac {\eta\, \left( \eta-1 \right) }{\nu}},\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&0\\ \noalign{\medskip}{\it j_{51}}&\mu\,{\it j_{42}}&{
\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array}
 \right] ,{\it det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}+{\it j_{42}}\,{\it
j_{53}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=1,\nu=0,\mu=\mu,
\varepsilon=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&\nu\,{\it j_{52}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it
j_{52}}&0\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,
{\it det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it
j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=\nu,\mu=0,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&-\mu\,{
\it j_{43}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\,\mu\,{\it j_{43}}\,{\it j_{62}}+{\it j_{43}}\,{
\it j_{51}}\,{\it j_{62}}-{\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]
\\ \noalign{\medskip}[\lambda=0,\eta=0,\nu=0,\mu=\mu,\varepsilon=0]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&0&0\end {array} \right] ,{\it det(j)}=-{
\it j_{43}}\,{\it j_{52}}\,{\it j_{61}}]\\ \noalign{\medskip}[\lambda=0,\eta=0,
\nu=0,\mu=-{\frac {{\it j_{53}}}{{\it j_{43}}}},\varepsilon=0]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\frac {
 \left( \eta-1 \right) {\it j_{42}}}{\nu}}&-{\frac {{\it j_{43}}\, \left(
\eta-1 \right) }{\nu}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it
j_{63}}\end {array} \right] ,{\it det(j)}=-{\frac { \left( {\it j_{42}}\,{\it
j_{63}}-{\it j_{43}}\,{\it j_{62}} \right)  \left( \eta\,{\it j_{41}}+{\it j_{51}}\,
\nu-{\it j_{41}} \right) }{\nu}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=1,\eta=\eta,\nu=\nu,\mu=-{\frac {\eta\, \left( \eta-1 \right)
}{\nu}}]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&\mu\,{\it
j_{42}}&\mu\,{\it j_{43}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}={\it j_{41}}\,\mu\,{\it j_{42}}\,{\it j_{63}}-
{\it j_{41}}\,\mu\,{\it j_{43}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{
\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=1,\eta=1,\nu=0,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{52}}\,\nu&{\it j_{53}}\,\nu\\ \noalign{\medskip}{\it j_{51}}&{
\it j_{52}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-{\it j_{52}}\,\nu\,{\it j_{51}}\,{\it j_{63}}
+{\it j_{53}}\,\nu\,{\it j_{51}}\,{\it j_{62}}+{\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-
{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda\neq 0,
\lambda=1,\eta=0,\nu=\nu,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=1,
\eta=0,\nu=0,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}=-{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}+{\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}
]\\ \noalign{\medskip}[\lambda\neq 0,\lambda=\lambda,\eta={\frac {\mu
\,{\it j_{43}}-{\it j_{53}}\,\lambda+{\it j_{53}}}{{\it j_{53}}}},\nu=-{\frac {{
\it j_{43}}\, \left( \mu\,{\it j_{43}}-2\,{\it j_{53}}\,\lambda+{\it j_{53}}
 \right) }{{{\it j_{53}}}^{2}}},\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&0\end {array} \right] ,{\it
det(j)}={\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\lambda
\neq 0,\lambda=\lambda,\eta=\lambda,\nu=\nu,\mu=0]\end {array}
 \right]
                              $$

		\end{enumerate}

\item CASE IV: $\mathfrak{k}=\mathfrak{r}_{3,\delta}$.
		\begin{enumerate}
			\item For a type (i) representation: there are 6 solutions.
			                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=-2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=i,\eta=2,\nu=i,\mu=0]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&0&0
\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}=2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]
\\ \noalign{\medskip}[\delta=-i,\eta=2,\nu=-i,\mu=0]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&i{\it j_{53}}
&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}-2
\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]\\ \noalign{\medskip}[\delta=i,\eta
=-2,\nu=i,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&-i{\it j_{43}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-i{\it
j_{53}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-2\,i{\it j_{41}}\,{\it j_{53}}\,{\it j_{63}}+
2\,i{\it j_{51}}\,{\it j_{43}}\,{\it j_{63}}]\\ \noalign{\medskip}[\delta=-i,
\eta=-2,\nu=-i,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\frac {i{\it j_{53}}\, \left( \eta+2 \right) }{\mu}}&{\frac {{
\it j_{53}}\, \left( \eta+2 \right) }{\mu}}\\ \noalign{\medskip}{\it j_{51}}
&i{\it j_{53}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&-i{\it j_{63}}&{\it
j_{63}}\end {array} \right] ,{\it det(j)}={\frac {-2\,i{\it j_{53}}\,{\it j_{63}}
\, \left( \eta\,{\it j_{51}}-\mu\,{\it j_{41}}+2\,{\it j_{51}} \right) }{\mu}}]
\\ \noalign{\medskip}[\delta=i,\eta=\eta,\nu=i,\mu=\mu]\end {array}
 \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\frac {-i{\it j_{53}}\, \left( \eta+2 \right) }{\mu}}&{\frac {{
\it j_{53}}\, \left( \eta+2 \right) }{\mu}}\\ \noalign{\medskip}{\it j_{51}}
&-i{\it j_{53}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&i{\it j_{63}}&{\it
j_{63}}\end {array} \right] ,{\it det(j)}={\frac {2\,i{\it j_{53}}\,{\it j_{63}}
\, \left( \eta\,{\it j_{51}}-\mu\,{\it j_{41}}+2\,{\it j_{51}} \right) }{\mu}}]
\\ \noalign{\medskip}[\delta=-i,\eta=\eta,\nu=-i,\mu=\mu]\end {array}
 \right]
                              $$

			\item For a type (ii) representation: there are 4 solutions.
			                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&-{\frac {
 \left( \eta-1 \right) {\it j_{42}}}{\nu}}&-{\frac { \left( \eta-1
 \right) {\it j_{43}}}{\nu}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{
\it j_{63}}\end {array} \right] ,{\it det(j)}=-{\frac { \left( {\it j_{42}}\,{
\it j_{63}}-{\it j_{43}}\,{\it j_{62}} \right)  \left( \eta\,{\it j_{41}}+\nu\,{
\it j_{51}}-{\it j_{41}} \right) }{\nu}}]\\ \noalign{\medskip}[\delta=0,\eta
=\eta,\nu=\nu,\mu=-{\frac {\eta\, \left( \eta-1 \right) }{\nu}}]
\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&{\it j_{42}}&{\it j_{43}}\\ \noalign{\medskip}{\it j_{51}}&\mu\,{\it
j_{42}}&\mu\,{\it j_{43}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}={\it j_{41}}\,\mu\,{\it j_{42}}\,{\it j_{63}}-
{\it j_{41}}\,\mu\,{\it j_{43}}\,{\it j_{62}}-{\it j_{42}}\,{\it j_{51}}\,{\it j_{63}}+{
\it j_{43}}\,{\it j_{51}}\,{\it j_{62}}]\\ \noalign{\medskip}[\delta=0,\eta=1,
\nu=0,\mu=\mu]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&\nu\,{\it j_{52}}&\nu\,{\it j_{53}}\\ \noalign{\medskip}{\it j_{51}}&{
\it j_{52}}&{\it j_{53}}\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}
\end {array} \right] ,{\it det(j)}=-\nu\,{\it j_{52}}\,{\it j_{51}}\,{\it j_{63}}
+\nu\,{\it j_{53}}\,{\it j_{51}}\,{\it j_{62}}+{\it j_{41}}\,{\it j_{52}}\,{\it j_{63}}-
{\it j_{41}}\,{\it j_{53}}\,{\it j_{62}}]\\ \noalign{\medskip}[\delta=0,\eta=0,
\nu=\nu,\mu=0]\end {array} \right]
                              $$
                              $$
 \left[ \begin {array}{c} [{\it j=}= \left[ \begin {array}{ccc} {
\it j_{41}}&0&0\\ \noalign{\medskip}{\it j_{51}}&{\it j_{52}}&{\it j_{53}}
\\ \noalign{\medskip}{\it j_{61}}&{\it j_{62}}&{\it j_{63}}\end {array}
 \right] ,{\it det(j)}={\it j_{41}}\, \left( {\it j_{52}}\,{\it j_{63}}-{\it j_{53}}
\,{\it j_{62}} \right) ]\\ \noalign{\medskip}[\delta=0,\eta=0,\nu=0,\mu=
\mu]\end {array} \right]
                              $$

		\end{enumerate}

\end{enumerate}

This gives the results in Table 2.

\subsection{Tables}

\

\pagebreak

Table 1 \quad \begin{tabular}{|c|c|c|c|}
\hline
\quad $\kk$ \quad & {\small Type} $t$ & {\small Existence results } & {\small \qquad Parameters of $t$  for the existence \qquad } \\ \hline
$\hh_1$ & {\small (i)} & {\small no} &  \\ \hline
 & {\small (ii)} & {\small no} &   \\ \hline
 & {\small (iii)} & {\small yes} & {\small $\nu=0$}\\ \hline
  & {\small (iv)} & {\small yes} & {\small $\eta=0$} \\ \hline
$\rr_3$ & {\small (i)} & {\small no } & \\ \hline
& {\small (ii)} & {\small no } & \\ \hline
& {\small (iii)} & {\small yes } & {\small $\nu=1$} \\ \hline
& {\small (iv)} & {\small yes } & {\small $\eta=1$} \\ \hline
$\rr_{3,\lambda}$ & {\small (i)} & {\small yes } & {\small $\eta=\nu=0, \mu=1, \varepsilon=0$}   \\
$\lambda=0$ & & & {\small $\eta=\mu, \nu=1, \varepsilon=0$} \\
& & & {\small $\eta=1, \nu=\mu=0, \varepsilon=0$}\\
& & & {\small $\nu=1, \mu=0, \varepsilon=0$}\\
& & & {\small $\nu=0, \mu=1, \varepsilon=0$} \\
& & & {\small $\eta=1, \nu=0, \varepsilon=0$} \\
& & & {\small $\eta=1, \mu=0, \varepsilon=0$} \\
& & & {\small $\eta=0, \mu=1, \varepsilon=0$}\\
& & & {\small $\eta=0, \nu=1, \varepsilon=0$} \\
& & & {\small $\eta=, \mu=\nu=1\varepsilon=0$} \\
& & & {\small $\eta=1, \nu=0, \mu=1,\varepsilon=0$} \\
& & & {\small $\eta=1, \nu=1, \mu=0, \varepsilon=0$} \\
 \hline
 & {\small (ii)} & {\small yes } &  {\small $\eta=1, \nu=\mu=\varepsilon=0$}\\
& & & {\small $\nu=1, \eta=\mu=\varepsilon=0$} \\
\hline
 & {\small (iii)} & {\small yes } & {\small $\eta=1, \nu=\varepsilon=0$}\\
 &  &  & {\small $\nu=1,\eta=\varepsilon=0$}\\
\hline
& {\small (iv)} & {\small no } & \\
\hline
$\rr_{3,\lambda}$,  & {\small (i)} & {\small yes } &  {\small $ \eta=1, \mu=\lambda$}\\
$\lambda\neq 0$ & & & {\small $\eta=1, \nu=\lambda$} \\
& & & {\small $\eta=\lambda, \nu=1$} \\
& & & {\small $\eta=\lambda, \mu=1$} \\
& & & {\small $\nu=1, \mu=\lambda$} \\
& & & {\small $\nu=\lambda, \mu=1$} \\
& & & {\small $\eta=1, \nu=\mu=\lambda$} \\
& & & {\small $\eta=\nu=1, \mu=\lambda$} \\
& & & {\small $\eta=\mu=1, \nu=\lambda$} \\
& & & {\small $\eta=\nu=\lambda, \mu=1$} \\
& & & {\small $\eta=\mu=\lambda, \nu=1$} \\
& & & {\small $\eta=\lambda, \nu=\mu=1$} \\
& & & {\small $\lambda=\eta=\nu=1$} \\
& & & {\small $\lambda=\eta=\mu=1$} \\
& & & {\small $\lambda=\nu=\mu=1$} \\
& & & {\small $\lambda=\eta=\nu=\mu=1$} \\
\hline
& {\small (ii)} & {\small yes } &  {\small $\lambda=\nu=1, \mu=0 $} \\
& & &  {\small $\eta=1, \nu=\lambda, \mu=0 $}\\
& & &  {\small $\eta=\lambda, \nu=1, \mu=0 $}\\
 \hline
& {\small (iii)} & {\small yes } &  {\small $\eta=1, \nu=\lambda$}\\
& & &  {\small $\eta=\lambda, \nu=1  $}\\
& & &  {\small $\lambda=\eta=\nu=1  $}\\
 \hline
& {\small (iv)} & {\small no } & \\
\hline
$\rr_{3,\delta}$ & {\small (i)} & {\small yes} & {\small $\delta=0, \nu=0, \mu=1$ }  \\
  & & & {\small $\delta=0, \eta=1, \nu=0 $} \\
\hline
  & {\small (ii)} & {\small yes} & {\small $\nu=1, \mu=\delta$} \\
& & & {\small $\nu=1, \mu=-\delta  $} \\
\hline
& {\small (iii)} & {\small yes} & {\small $\delta=0, \eta=\nu=1$}\\
\hline
& {\small (iv)} & {\small no} & \\
\hline
\end{tabular}

\pagebreak

Table 2
 \quad \begin{tabular}{|c|c|c|c|}
\hline
\quad $\kk$ \quad & {\small Type} $t$ & {\small Existence results } & {\small \qquad Parameters of $t$  for the existence \qquad } \\ \hline
$\hh_1$ & {\small (i)} & {\small yes} & {\small $\nu\neq 0, \mu=-\frac{\eta^{2}}{\nu}$} \\
& & & {\small $\eta=\nu=0$} \\ \hline
 & {\small (ii)} & {\small no} &   \\ \hline
$\rr_3$ & {\small (i)} & {\small no } & \\ \hline
$\rr_3$ & {\small (ii)} & {\small no } & \\
 \hline
$\rr_{3,\lambda}$ & {\small (i)} & {\small yes } & {\small $\lambda=0, \nu\neq 0, \varepsilon=0$}\\
& & & {\small $\lambda=0, \eta=1, \nu=0, \varepsilon =0 $} \\
& & & {\small $\lambda=0, \eta=-1, \nu=0, \varepsilon=0 $}  \\
& & & {\small $\lambda=-1, \eta=1, \mu=0 $}\\
& & & {\small $\lambda=-1, \eta=\frac{j_{43}\mu + j_{53}}{j_{53}^{2}}, \nu=-\frac{(j_{43}\mu+2j_{53})j_{43}}{j_{53}^{2}}$}\\
& & & {\small $\lambda=-1, \eta=-1, \nu=2\frac{j_{42}}{j_{52}}, \mu=0 $} \\
 \hline
& {\small (ii)} & {\small yes } &  {\small $ \lambda=0, \nu\neq 0, \mu=-\frac{\eta(\eta-1)}{\nu}, \varepsilon=0$}\\
& & & {\small $ \lambda=0, \eta=1, \nu=0, \varepsilon=0$} \\
& & & {\small $ \lambda=0, \eta=\nu=0, \varepsilon=0$} \\
& & & {\small $ \lambda=0, \eta=\mu=0, \varepsilon=0$} \\
& & & {\small $ \lambda=0, \eta=\nu=0, \mu=-\frac{j_{53}}{j_{43}}, \varepsilon=0$} \\
& & & {\small $ \lambda=1, \nu\neq 0, \mu=-\frac{\eta(\eta-1)}{\nu}$} \\
& & & {\small $ \lambda=1, \eta=1, \nu=0$}\\
& & & {\small $ \lambda=1, \eta=\mu=0$} \\
& & & {\small $\lambda=1, \eta=\nu=0$} \\
& & & {\small $\lambda\neq 0, \eta=\frac{\mu j_{43}-\lambda j_{53}+j_{53}}{j_{53}}, \nu=-\frac{j_{43}(\mu j_{43}-2\lambda j_{53}+j_{53})}{j_{53}^{2}} $} \\
& & & {\small $\lambda\neq 0, \eta=\lambda, \mu=0$}\\
\hline
$\rr_{3,\delta}$ & {\small (i)} & {\small no} & {\small $$ }  \\
  \hline
  & {\small (ii)} & {\small yes} & {\small $\delta=0, \nu\neq 0, \mu=-\frac{\eta(\eta-1)}{\nu}$} \\
& & & {\small $\delta=0, \eta=1, \nu=0$} \\
& & & {\small $\delta=0, \eta=\mu=0 $} \\
& & & {\small $\delta=0, \eta=\nu=0$}\\
\hline
\end{tabular}


TITLE: Local minima of a nonlinear optimization problem
QUESTION [0 upvotes]: Consider the following non-linear optimization problem:
$${min}_c\left(\sum_{i=1}^nc_ix_i-7\right)^2$$

How can we determine the number of local minima points that this optimization problem has?

Is there also a way to find them?

REPLY [0 votes]: The problem is convex and therefore has a unique minimum value. All $c$ that satisfy $\sum_{i=1}^n c_i x_i=7$ (obtained from setting the derivative to $0$) solve the optimization problem. This is a linear system that has either $0$, $1$ or $\infty$ solutions.


TITLE: Drude model: A concerning approach to $\tau$ hurts Ohm's Law
QUESTION [2 upvotes]: Here's the question. Between two points, $a$ and $b$, if an object is accelerated between them, it will cross the distance between them in time $t=\sqrt{2\frac{d}{a}}$. So, if the acceleration is increased, the time decreases with the square root of $a$.
If we take the average $t$ for sinusoidally distributed $d$ values, our $\tau$ should be $\tau = \frac{t}{\sqrt{2}}$. Therefore, increasing the $E$ field should decrease the value of $\tau$ by the square root of the increase. Therefore, the current density equation $j=nq^2\frac{\tau}{m}E$ should be scaling up $j$ by the square root of $E$ since we substitute $\tau=\sqrt{\frac{d}{E}}$. This defeats Ohm's Law. I don't want it to though because its seemingly wrong, but I don't know why.

REPLY [4 votes]: The electric field (or whatever force) has only a very small effect on the time between collisions.  The reason is that electrons are very light, and their thermal velocity exceeds any change in the velocity due to the applied fields.
One thus treats the scattering time (tau) as a constant.
The applied electric field then causes a drift velocity, tau*acceleration.
Oh, it's assumed that every collision resets the electron velocity randomly... it doesn't remember it's previous velocity.


TITLE: Why is $\frac{n!}{n^n} \leq \frac{2}{n^2}$
QUESTION [1 upvotes]: I was reading the solution to an example problem for the  Comparison Test for Series regarding the following series:
$$\large \sum_{n = 1}\frac{n!}{n^n}$$
I understood every part of the answer except when he stated that
$$\large \frac{n!}{n^n} \leq \frac{2}{n^2}$$
The purpose of this was to find a $\large b_n$ such that $\large a_n \leq b_n $ for the Comparison Test, but I just couldn't grasp the way he justified it.
He explained it as follows:
If you write out the expression $\frac{n!}{n^n}$ in the general form you get:
$$\large \frac{n!}{n^n} = \frac{n}{n}\frac{(n-1)}{n}\frac{(n-2)}{n}....\frac{2}{n}\frac{1}{n}$$
Which makes perfect sennse. He then went on to explain how every term past $\large \frac{n}{n}$ would be less than $1$ which also makes sense, but then he said because of that we could state that
$$\frac{n!}{n^n} \leq \frac{2}{n}\frac{1}{n}$$ or $$\frac{n!}{n^n} \leq \frac{2}{n^2}$$ and could therefore use $\large b_n = \frac{2}{n^2}$ for the Comparison Test.
I just couldn't follow his logic when stating that $\large \frac{n!}{n^n} \leq \frac{2}{n^2}$. I believe it, but he said it in such a way as to imply that it was intuitive; that the conclusion could easily be stated given the presented information but it just didn't click for me. Could someone explain in greater detail why that relationship should be forthcoming?

REPLY [3 votes]: Perhaps look at it from a somewhat informal way, take $n=5$
We have to show $\frac{5!}{5^5}\leq \frac{2}{5^2}$ Now write out: $\frac{5.4.3.2.1}{5.5.5.5.5} \leq \frac{5.5.5.2.1}{5.5.5.5.5}$ That last fraction becomes $\frac{2}{5^2}$. Maybe you can see the general case for $n$ now


TITLE: Dugundji topology textbook. Is this an error?
QUESTION [1 upvotes]: 3.4. Theorem
  Let $W$ be well-ordered, and $\Sigma\subset I(W)$ any family with the following properties:
  (a) Each union of members of $\Sigma$ belongs to $\Sigma$.
  (b) If $W(a)\in\Sigma$, then also $W(a)\cup \{a\}\in\Sigma$.
  Then $\Sigma = I(W)$ and, in particular, $W\in\Sigma$.

$I(W)$ denotes the set of all ideals of $W$.
But isn't the following a counterexample to this Theorem?  Consider $W = \{1,2,3,4,5\}$ and $\Sigma=\{\{1,2,3,4\},\{1,2,3,4,5\}\}$. These meet the conditions (a), (b), correct?
But $\Sigma\neq I(W).$

REPLY [1 votes]: Your $\Sigma$ does not satisfy condition (a): if you take the subset $\emptyset\subset\Sigma$, then the union $\bigcup\emptyset=\emptyset$ is not an element of $\Sigma$.  That is, the union of elements of $\Sigma$ consisting of no elements at all is not in $\Sigma$.


TITLE: Don´t know how to start proving this formula.
QUESTION [0 upvotes]: \begin{equation*}
\int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}-
\frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1).
\end{equation*}
I`d like to know if someone could help me with this exercise. Maybe there´s a hint I don´t see right now.

REPLY [0 votes]: I guess the reduction formula should be like this:
$$
\begin{aligned}
I(m, n) :&=\int \frac{\cos ^{m} x}{\sin ^{n} x} d x \\&=-\frac{1}{n-1} \int \cos ^{m-1} x d\left(\frac{1}{\sin ^{n-1} x}\right) \\
&=-\frac{1}{n-1}\left(\frac{\cos ^{m-1} x}{\sin ^{n-1} x}+\int \frac{(m-1) \cos ^{m-2} x}{\sin ^{n-2} x} d x\right) \\
&=-\frac{\cos^{m-1} x}{(n-1) \sin ^{n-1}x}-\frac{m-1}{n-1} I(m-2, n-2)
\end{aligned}
$$


TITLE: Solving nonlinear differential equation by substitution
QUESTION [0 upvotes]: Has someone an idea, how to get a differential equation for $\Phi (x) := ||y(x)||^2$ from this nonlinear ODE?
$\begin{pmatrix}
y'_1(x)\\ y'_2(x)
\end{pmatrix}
=
\begin{pmatrix}
0 & -1\\ 1 & 0
\end{pmatrix}
\begin{pmatrix}
y_1(x)\\ y_2(x)
\end{pmatrix}
+ \epsilon||y(x)||^2
\begin{pmatrix}
y_1(x)\\ y_2(x)
\end{pmatrix}$?
Thanks & have a nice day!

REPLY [0 votes]: Or simply switch to polar coordinates
$$x = r \cos{\theta}$$
$$y = r \sin{\theta}$$
Then your equation simplifies significantly and you get the equation that you need plus you solve the system completely.


TITLE: If $T= T_1 \cup T_2$ is inconsistent, then can we conclude that $T_1 \models \neg \phi$ for some $\phi\in T_2$
QUESTION [2 upvotes]: I have seen this reasoning elsewhere and I am not sure wether it is correct.
I think the reasoning behind it should be as follows: Suppose that T is inconsistent.
Then $T_2$ is inconsistent with $T_1$ and so there is $\phi \in T_2$ such that $T_1 \nvDash \phi$.
But I can't find a proof of why it is the case that $T_2$ is inconsistent with $T_1$, as I only know that they both should be inconsistent by compactness.
Also, we need to know that $T_1$ is complete to deduce to know that $T_1 \models \neg \phi$.
What is missing in this argument?

REPLY [0 votes]: Suppose that $T=T_1 \cup T_2$ is inconsistent, and $T, T_1, T_2 \neq \emptyset$ then by compactness, it is not the case that every finite subset of T is consistent.
As $T_1, T_2 \subseteq T$, w.l.o.g suppose that there is M such that $M \models T_1$, then $M \nvDash T_2$ so there is some $\phi \in T_2$ such that $T_1 \nvDash \phi$.
If additionally $T_1$ is complete: $T_1 \models \neg \phi$.


TITLE: How do I efficiently budget for digits when converting from one base to another?
QUESTION [1 upvotes]: I'm trying to work out a formula that gives me the minimum number of digits required to express a number in a target radix (r2) knowing only the source radix (r1) and number of significant digits (d).
I've been messing around with this for quite some time but I think I've figured out that the formula is:
$\left\lfloor \log_{r2}(r1) × d\right\rfloor + 1$
My question is, did I get this right? It works for a variety of examples, but I know that doesn't necessarily mean it's correct. I lack the skills to prove it. I didn't study advanced math. Please go easy on me.

REPLY [1 votes]: Let $r_1, r_2 > 1$ be the two bases, then the smallest number requiring $n+1$ digits in base $r_1$ is $r_1^n$ which has representation $10\ldots 0$ (where we have $n$ zeros), therefore an arbitrary number $m$ needs $n+1$ digits if and only if
$$
 r_1^n \le m < r_1^{n+1}
$$
and this relation holds in any base. Also for the same $m$ we have
$$
 r_2^{n \cdot \log_{r_2} r_1} \le m < r_2^{(n+1) \cdot \log_{r_2} r_1}.
$$
So the number of digits needed to represent $m$ in base $r_2$ is contained between $\lfloor n\log_{r_2} r_1 \rfloor + 1, \ldots, \lceil (n+1)\log_{r_2} r_1 \rceil + 1$. So your formula is indeed a good approximation and some times it is right. For example consider $r_1 = 2, r_2 = 3$, and:
$$
 \begin{array}{l|l|l|l|c|c}
   m & r_1 & r_2 & d_2 & d_3 & \lfloor (d_2-1)\cdot \log_3 2 \rfloor + 1 & \lceil d_2\cdot \log_3 2\rceil + 1 \\ \hline
   0 & 0 & 0 & 1 & 1 & 1 & 2 \\
   1 & 1 & 1 & 1 & 1 & 1 & 2 \\
   2 & 10 & 2 & 2 & 1 & 1 & 3 \\
   3 & 11 & 10 & 2 & 2 & 1 & 3 \\
   4 & 100 & 11 & 3 & 2 & 2 & 3 \\
   5 & 101 & 12 & 3 & 2 & 2 & 3 \\
   6 & 110 & 100 & 3 & 3 & 2 & 3 \\
   7 & 111 & 101 & 3 & 3 & 2 & 3 \\
 \end{array}
$$
As you can see in the table, for $m = 3$ the number of digits needed in base $3$ is strictly between the bounds, for $m \in \{0,1,2,4,5\}$ the lower bound is assumed, and for $m \in \{ 6,7 \}$ the upper bound is assumed.