\documentclass{article}


\usepackage{amsmath,amssymb}


\sloppy


\begin{document}


This output from the file \texttt{CSsl22.red}.\\


Computation of all complex    structures on the real Lie Algebra


USD {\mathcal{G}}_{6,sl2bisxsl2bis}.USD


\smallskip  \par 


Commutation relations for


USD {\mathcal{G}}_{6,sl2bisxsl2bis}:USD\\


USD[x(1),x(2)]=x(3)USD;


USD[x(1),x(3)]=x(2)USD;


USD[x(2),x(3)]=x(1)USD;


USD[x(4),x(5)]=x(6)USD;


USD[x(4),x(6)]=x(5)USD;


USD[x(5),x(6)]=x(4)USD;


\P


Nonzero torsion


\par


Torsion equations to cancel (Latex output) : \\USD


{1,2}|1\\(xi(2,2) + xi(1,1))*xi(1,3) + (xi(2,2) - xi(1,1))*xi(3,1) - (xi(2,1) - 
xi(1,2))*xi(3,2)\\


{1,2}|2\\(xi(2,2) + xi(1,1))*xi(2,3) - (xi(2,1) - xi(1,2))*xi(3,1) + (xi(2,2) - 
xi(1,1))*xi(3,2)\\


{1,2}|3\\xi(2,1)*xi(1,2) + 1 - xi(2,2)*xi(1,1) - xi(3,1)**2 + xi(3,2)**2 + (xi(2
,2) + xi(1,1))*xi(3,3)\\


{1,2}|4\\xi(4,2)*xi(3,2) - xi(4,1)*xi(3,1) + (xi(2,2) + xi(1,1))*xi(4,3) + xi(6,
1)*xi(5,2) - xi(6,2)*xi(5,1)\\


{1,2}|5\\xi(5,2)*xi(3,2) - xi(5,1)*xi(3,1) + (xi(2,2) + xi(1,1))*xi(5,3) + xi(6,
1)*xi(4,2) - xi(6,2)*xi(4,1)\\


{1,2}|6\\ - (xi(5,2)*xi(4,1) - xi(5,1)*xi(4,2) + xi(6,1)*xi(3,1) - xi(6,2)*xi(3,
2)) + (xi(2,2) + xi(1,1))*xi(6,3)\\


{1,3}|1\\(xi(2,1) + xi(1,2))*xi(1,1) + xi(2,3)*xi(1,3) + xi(3,1)*xi(2,3) - (xi(2
,1) - xi(1,2))*xi(3,3)\\


{1,3}|2\\xi(2,1)**2 + 1 + xi(2,2)*xi(1,1) + xi(2,3)**2 + xi(3,1)*xi(1,3) + (xi(2
,2) - xi(1,1))*xi(3,3)\\


{1,3}|3\\ - (xi(2,3)*xi(1,1) - xi(2,1)*xi(1,3) - xi(3,1)*xi(2,1) - xi(3,2)*xi(1,
1)) + (xi(3,2) + xi(2,3))*xi(3,3)\\


{1,3}|4\\xi(4,2)*xi(3,3) + xi(4,2)*xi(1,1) + xi(4,1)*xi(2,1) + xi(4,3)*xi(2,3) +
 xi(6,1)*xi(5,3) - xi(6,3)*xi(5,1)\\


{1,3}|5\\xi(5,2)*xi(3,3) + xi(5,2)*xi(1,1) + xi(5,1)*xi(2,1) + xi(5,3)*xi(2,3) +
 xi(6,1)*xi(4,3) - xi(6,3)*xi(4,1)\\


{1,3}|6\\ - (xi(5,3)*xi(4,1) - xi(5,1)*xi(4,3) - xi(6,1)*xi(2,1) - (xi(3,3) + xi
(1,1))*xi(6,2)) + xi(6,3)*xi(2,3)\\


{1,4}|1\\(xi(3,1) + xi(1,3))*xi(2,4) - (xi(2,1) - xi(1,2))*xi(3,4) - xi(5,1)*xi(
1,6) - xi(6,1)*xi(1,5)\\


{1,4}|2\\xi(3,1)*xi(1,4) + xi(2,4)*xi(2,3) + (xi(2,2) - xi(1,1))*xi(3,4) - xi(5,
1)*xi(2,6) - xi(6,1)*xi(2,5)\\


{1,4}|3\\ - (xi(2,4)*xi(1,1) - xi(2,1)*xi(1,4) - xi(3,3)*xi(2,4) - xi(3,4)*xi(3,
2) + xi(5,1)*xi(3,6)) - xi(6,1)*xi(3,5)\\


{1,4}|4\\xi(4,3)*xi(2,4) + xi(4,2)*xi(3,4) - xi(5,1)*xi(4,6) + (xi(5,4) - xi(4,5
))*xi(6,1) - xi(6,4)*xi(5,1)\\


{1,4}|5\\xi(5,3)*xi(2,4) + xi(5,2)*xi(3,4) - xi(5,6)*xi(5,1) - (xi(5,5) - xi(4,4
))*xi(6,1) - xi(6,4)*xi(4,1)\\


{1,4}|6\\ - (xi(5,4)*xi(4,1) - xi(5,1)*xi(4,4) - xi(6,2)*xi(3,4) - xi(6,3)*xi(2,
4) + xi(6,5)*xi(6,1)) - xi(6,6)*xi(5,1)\\


{1,5}|1\\(xi(3,1) + xi(1,3))*xi(2,5) - (xi(2,1) - xi(1,2))*xi(3,5) + xi(4,1)*xi(
1,6) - xi(6,1)*xi(1,4)\\


{1,5}|2\\xi(3,1)*xi(1,5) + xi(2,5)*xi(2,3) + (xi(2,2) - xi(1,1))*xi(3,5) + xi(4,
1)*xi(2,6) - xi(6,1)*xi(2,4)\\


{1,5}|3\\ - (xi(2,5)*xi(1,1) - xi(2,1)*xi(1,5) - xi(3,3)*xi(2,5) - xi(3,5)*xi(3,
2) - xi(4,1)*xi(3,6)) - xi(6,1)*xi(3,4)\\


{1,5}|4\\xi(4,3)*xi(2,5) + xi(4,2)*xi(3,5) + xi(4,6)*xi(4,1) + (xi(5,5) - xi(4,4
))*xi(6,1) - xi(6,5)*xi(5,1)\\


{1,5}|5\\xi(5,3)*xi(2,5) + xi(5,2)*xi(3,5) + xi(5,6)*xi(4,1) - (xi(5,4) - xi(4,5
))*xi(6,1) - xi(6,5)*xi(4,1)\\


{1,5}|6\\ - (xi(5,5)*xi(4,1) - xi(5,1)*xi(4,5) - xi(6,2)*xi(3,5) - xi(6,3)*xi(2,
5) + xi(6,4)*xi(6,1)) + xi(6,6)*xi(4,1)\\


{1,6}|1\\(xi(3,1) + xi(1,3))*xi(2,6) - (xi(2,1) - xi(1,2))*xi(3,6) + xi(4,1)*xi(
1,5) + xi(5,1)*xi(1,4)\\


{1,6}|2\\xi(3,1)*xi(1,6) + xi(2,6)*xi(2,3) + (xi(2,2) - xi(1,1))*xi(3,6) + xi(4,
1)*xi(2,5) + xi(5,1)*xi(2,4)\\


{1,6}|3\\ - (xi(2,6)*xi(1,1) - xi(2,1)*xi(1,6) - xi(3,3)*xi(2,6) - xi(3,6)*xi(3,
2) - xi(4,1)*xi(3,5)) + xi(5,1)*xi(3,4)\\


{1,6}|4\\xi(4,3)*xi(2,6) + xi(4,2)*xi(3,6) + xi(4,5)*xi(4,1) + xi(5,1)*xi(4,4) +
 xi(6,1)*xi(5,6) - xi(6,6)*xi(5,1)\\


{1,6}|5\\xi(5,3)*xi(2,6) + xi(5,2)*xi(3,6) + xi(5,4)*xi(5,1) + xi(5,5)*xi(4,1) +
 xi(6,1)*xi(4,6) - xi(6,6)*xi(4,1)\\


{1,6}|6\\ - (xi(5,6)*xi(4,1) - xi(5,1)*xi(4,6) - xi(6,2)*xi(3,6) - xi(6,3)*xi(2,
6) - xi(6,4)*xi(5,1)) + xi(6,5)*xi(4,1)\\


{2,3}|1\\xi(1,2)**2 + 1 - xi(1,3)**2 + xi(2,2)*xi(1,1) + xi(3,2)*xi(2,3) - (xi(2
,2) - xi(1,1))*xi(3,3)\\


{2,3}|2\\(xi(2,1) + xi(1,2))*xi(2,2) - xi(2,3)*xi(1,3) + xi(3,2)*xi(1,3) + (xi(2
,1) - xi(1,2))*xi(3,3)\\


{2,3}|3\\ - (xi(2,3)*xi(1,2) - xi(2,2)*xi(1,3) - xi(3,1)*xi(2,2) - xi(3,2)*xi(1,
2)) + (xi(3,1) - xi(1,3))*xi(3,3)\\


{2,3}|4\\(xi(3,3) + xi(2,2))*xi(4,1) + xi(4,2)*xi(1,2) - xi(4,3)*xi(1,3) + xi(6,
2)*xi(5,3) - xi(6,3)*xi(5,2)\\


{2,3}|5\\(xi(3,3) + xi(2,2))*xi(5,1) + xi(5,2)*xi(1,2) - xi(5,3)*xi(1,3) + xi(6,
2)*xi(4,3) - xi(6,3)*xi(4,2)\\


{2,3}|6\\ - (xi(5,3)*xi(4,2) - xi(5,2)*xi(4,3) - (xi(3,3) + xi(2,2))*xi(6,1) - 
xi(6,2)*xi(1,2)) - xi(6,3)*xi(1,3)\\


{2,4}|1\\xi(3,2)*xi(2,4) - xi(1,4)*xi(1,3) - (xi(2,2) - xi(1,1))*xi(3,4) - xi(5,
2)*xi(1,6) - xi(6,2)*xi(1,5)\\


{2,4}|2\\(xi(3,2) - xi(2,3))*xi(1,4) + (xi(2,1) - xi(1,2))*xi(3,4) - xi(5,2)*xi(
2,6) - xi(6,2)*xi(2,5)\\


{2,4}|3\\ - (xi(2,4)*xi(1,2) - xi(2,2)*xi(1,4) + xi(3,3)*xi(1,4) - xi(3,4)*xi(3,
1) + xi(5,2)*xi(3,6)) - xi(6,2)*xi(3,5)\\


{2,4}|4\\ - (xi(4,3)*xi(1,4) - xi(4,1)*xi(3,4) + xi(5,2)*xi(4,6) - (xi(5,4) - xi
(4,5))*xi(6,2)) - xi(6,4)*xi(5,2)\\


{2,4}|5\\ - (xi(5,3)*xi(1,4) - xi(5,1)*xi(3,4) + xi(5,6)*xi(5,2) + (xi(5,5) - xi
(4,4))*xi(6,2)) - xi(6,4)*xi(4,2)\\


{2,4}|6\\ - (xi(5,4)*xi(4,2) - xi(5,2)*xi(4,4) - xi(6,1)*xi(3,4) + xi(6,3)*xi(1,
4) + xi(6,5)*xi(6,2)) - xi(6,6)*xi(5,2)\\


{2,5}|1\\xi(3,2)*xi(2,5) - xi(1,5)*xi(1,3) - (xi(2,2) - xi(1,1))*xi(3,5) + xi(4,
2)*xi(1,6) - xi(6,2)*xi(1,4)\\


{2,5}|2\\(xi(3,2) - xi(2,3))*xi(1,5) + (xi(2,1) - xi(1,2))*xi(3,5) + xi(4,2)*xi(
2,6) - xi(6,2)*xi(2,4)\\


{2,5}|3\\ - (xi(2,5)*xi(1,2) - xi(2,2)*xi(1,5) + xi(3,3)*xi(1,5) - xi(3,5)*xi(3,
1) - xi(4,2)*xi(3,6)) - xi(6,2)*xi(3,4)\\


{2,5}|4\\ - (xi(4,3)*xi(1,5) - xi(4,1)*xi(3,5) - xi(4,6)*xi(4,2) - (xi(5,5) - xi
(4,4))*xi(6,2)) - xi(6,5)*xi(5,2)\\


{2,5}|5\\ - (xi(5,3)*xi(1,5) - xi(5,1)*xi(3,5) - xi(5,6)*xi(4,2) + (xi(5,4) - xi
(4,5))*xi(6,2)) - xi(6,5)*xi(4,2)\\


{2,5}|6\\ - (xi(5,5)*xi(4,2) - xi(5,2)*xi(4,5) - xi(6,1)*xi(3,5) + xi(6,3)*xi(1,
5) + xi(6,4)*xi(6,2)) + xi(6,6)*xi(4,2)\\


{2,6}|1\\xi(3,2)*xi(2,6) - xi(1,6)*xi(1,3) - (xi(2,2) - xi(1,1))*xi(3,6) + xi(4,
2)*xi(1,5) + xi(5,2)*xi(1,4)\\


{2,6}|2\\(xi(3,2) - xi(2,3))*xi(1,6) + (xi(2,1) - xi(1,2))*xi(3,6) + xi(4,2)*xi(
2,5) + xi(5,2)*xi(2,4)\\


{2,6}|3\\ - (xi(2,6)*xi(1,2) - xi(2,2)*xi(1,6) + xi(3,3)*xi(1,6) - xi(3,6)*xi(3,
1) - xi(4,2)*xi(3,5)) + xi(5,2)*xi(3,4)\\


{2,6}|4\\ - (xi(4,3)*xi(1,6) - xi(4,1)*xi(3,6) - xi(4,5)*xi(4,2) - xi(5,2)*xi(4,
4) - xi(6,2)*xi(5,6)) - xi(6,6)*xi(5,2)\\


{2,6}|5\\ - (xi(5,3)*xi(1,6) - xi(5,1)*xi(3,6) - xi(5,4)*xi(5,2) - xi(5,5)*xi(4,
2) - xi(6,2)*xi(4,6)) - xi(6,6)*xi(4,2)\\


{2,6}|6\\ - (xi(5,6)*xi(4,2) - xi(5,2)*xi(4,6) - xi(6,1)*xi(3,6) + xi(6,3)*xi(1,
6) - xi(6,4)*xi(5,2)) + xi(6,5)*xi(4,2)\\


{3,4}|1\\ - (xi(2,4)*xi(1,1) + xi(1,4)*xi(1,2) - xi(3,3)*xi(2,4) + xi(3,4)*xi(2,
3) + xi(5,3)*xi(1,6)) - xi(6,3)*xi(1,5)\\


{3,4}|2\\ - (xi(2,4)*xi(2,1) + xi(2,2)*xi(1,4) - xi(3,3)*xi(1,4) + xi(3,4)*xi(1,
3) + xi(5,3)*xi(2,6)) - xi(6,3)*xi(2,5)\\


{3,4}|3\\ - (xi(2,4)*xi(1,3) - xi(2,3)*xi(1,4) + xi(3,1)*xi(2,4) + xi(3,2)*xi(1,
4) + xi(5,3)*xi(3,6)) - xi(6,3)*xi(3,5)\\


{3,4}|4\\ - (xi(4,2)*xi(1,4) + xi(4,1)*xi(2,4) + xi(5,3)*xi(4,6) - (xi(5,4) - xi
(4,5))*xi(6,3)) - xi(6,4)*xi(5,3)\\


{3,4}|5\\ - (xi(5,2)*xi(1,4) + xi(5,1)*xi(2,4) + xi(5,6)*xi(5,3) + (xi(5,5) - xi
(4,4))*xi(6,3)) - xi(6,4)*xi(4,3)\\


{3,4}|6\\ - (xi(5,4)*xi(4,3) - xi(5,3)*xi(4,4) + xi(6,1)*xi(2,4) + xi(6,2)*xi(1,
4) + xi(6,5)*xi(6,3)) - xi(6,6)*xi(5,3)\\


{3,5}|1\\ - (xi(2,5)*xi(1,1) + xi(1,5)*xi(1,2) - xi(3,3)*xi(2,5) + xi(3,5)*xi(2,
3) - xi(4,3)*xi(1,6)) - xi(6,3)*xi(1,4)\\


{3,5}|2\\ - (xi(2,5)*xi(2,1) + xi(2,2)*xi(1,5) - xi(3,3)*xi(1,5) + xi(3,5)*xi(1,
3) - xi(4,3)*xi(2,6)) - xi(6,3)*xi(2,4)\\


{3,5}|3\\ - (xi(2,5)*xi(1,3) - xi(2,3)*xi(1,5) + xi(3,1)*xi(2,5) + xi(3,2)*xi(1,
5) - xi(4,3)*xi(3,6)) - xi(6,3)*xi(3,4)\\


{3,5}|4\\ - (xi(4,2)*xi(1,5) + xi(4,1)*xi(2,5) - xi(4,6)*xi(4,3) - (xi(5,5) - xi
(4,4))*xi(6,3)) - xi(6,5)*xi(5,3)\\


{3,5}|5\\ - (xi(5,2)*xi(1,5) + xi(5,1)*xi(2,5) - xi(5,6)*xi(4,3) + (xi(5,4) - xi
(4,5))*xi(6,3)) - xi(6,5)*xi(4,3)\\


{3,5}|6\\ - (xi(5,5)*xi(4,3) - xi(5,3)*xi(4,5) + xi(6,1)*xi(2,5) + xi(6,2)*xi(1,
5) + xi(6,4)*xi(6,3)) + xi(6,6)*xi(4,3)\\


{3,6}|1\\ - (xi(2,6)*xi(1,1) + xi(1,6)*xi(1,2) - xi(3,3)*xi(2,6) + xi(3,6)*xi(2,
3) - xi(4,3)*xi(1,5)) + xi(5,3)*xi(1,4)\\


{3,6}|2\\ - (xi(2,6)*xi(2,1) + xi(2,2)*xi(1,6) - xi(3,3)*xi(1,6) + xi(3,6)*xi(1,
3) - xi(4,3)*xi(2,5)) + xi(5,3)*xi(2,4)\\


{3,6}|3\\ - (xi(2,6)*xi(1,3) - xi(2,3)*xi(1,6) + xi(3,1)*xi(2,6) + xi(3,2)*xi(1,
6) - xi(4,3)*xi(3,5)) + xi(5,3)*xi(3,4)\\


{3,6}|4\\ - (xi(4,2)*xi(1,6) + xi(4,1)*xi(2,6) - xi(4,5)*xi(4,3) - xi(5,3)*xi(4,
4) - xi(6,3)*xi(5,6)) - xi(6,6)*xi(5,3)\\


{3,6}|5\\ - (xi(5,2)*xi(1,6) + xi(5,1)*xi(2,6) - xi(5,4)*xi(5,3) - xi(5,5)*xi(4,
3) - xi(6,3)*xi(4,6)) - xi(6,6)*xi(4,3)\\


{3,6}|6\\ - (xi(5,6)*xi(4,3) - xi(5,3)*xi(4,6) + xi(6,1)*xi(2,6) + xi(6,2)*xi(1,
6) - xi(6,4)*xi(5,3)) + xi(6,5)*xi(4,3)\\


{4,5}|1\\ - (xi(3,5)*xi(2,4) - xi(3,4)*xi(2,5) - xi(4,4)*xi(1,6) - xi(5,5)*xi(1,
6) + xi(6,4)*xi(1,4)) + xi(6,5)*xi(1,5)\\


{4,5}|2\\ - (xi(3,5)*xi(1,4) - xi(3,4)*xi(1,5) - xi(4,4)*xi(2,6) - xi(5,5)*xi(2,
6) + xi(6,4)*xi(2,4)) + xi(6,5)*xi(2,5)\\


{4,5}|3\\ - (xi(2,5)*xi(1,4) - xi(2,4)*xi(1,5) - xi(4,4)*xi(3,6) - xi(5,5)*xi(3,
6) + xi(6,4)*xi(3,4)) + xi(6,5)*xi(3,5)\\


{4,5}|4\\(xi(5,5) + xi(4,4))*xi(4,6) + (xi(5,5) - xi(4,4))*xi(6,4) - (xi(5,4) - 
xi(4,5))*xi(6,5)\\


{4,5}|5\\(xi(5,5) + xi(4,4))*xi(5,6) - (xi(5,4) - xi(4,5))*xi(6,4) + (xi(5,5) - 
xi(4,4))*xi(6,5)\\


{4,5}|6\\xi(5,4)*xi(4,5) + 1 - xi(5,5)*xi(4,4) - xi(6,4)**2 + xi(6,5)**2 + (xi(5
,5) + xi(4,4))*xi(6,6)\\


{4,6}|1\\ - (xi(3,6)*xi(2,4) - xi(3,4)*xi(2,6) - xi(4,4)*xi(1,5) - xi(5,4)*xi(1,
4) - xi(5,6)*xi(1,6)) + xi(6,6)*xi(1,5)\\


{4,6}|2\\ - (xi(3,6)*xi(1,4) - xi(3,4)*xi(1,6) - xi(4,4)*xi(2,5) - xi(5,4)*xi(2,
4) - xi(5,6)*xi(2,6)) + xi(6,6)*xi(2,5)\\


{4,6}|3\\ - (xi(2,6)*xi(1,4) - xi(2,4)*xi(1,6) - xi(4,4)*xi(3,5) - xi(5,4)*xi(3,
4) - xi(5,6)*xi(3,6)) + xi(6,6)*xi(3,5)\\


{4,6}|4\\(xi(5,4) + xi(4,5))*xi(4,4) + xi(5,6)*xi(4,6) + xi(6,4)*xi(5,6) - (xi(5
,4) - xi(4,5))*xi(6,6)\\


{4,6}|5\\xi(5,4)**2 + 1 + xi(5,5)*xi(4,4) + xi(5,6)**2 + xi(6,4)*xi(4,6) + (xi(5
,5) - xi(4,4))*xi(6,6)\\


{4,6}|6\\ - (xi(5,6)*xi(4,4) - xi(5,4)*xi(4,6) - xi(6,4)*xi(5,4) - xi(6,5)*xi(4,
4)) + (xi(6,5) + xi(5,6))*xi(6,6)\\


{5,6}|1\\ - (xi(3,6)*xi(2,5) - xi(3,5)*xi(2,6) - xi(4,5)*xi(1,5) + xi(4,6)*xi(1,
6) - xi(5,5)*xi(1,4)) + xi(6,6)*xi(1,4)\\


{5,6}|2\\ - (xi(3,6)*xi(1,5) - xi(3,5)*xi(1,6) - xi(4,5)*xi(2,5) + xi(4,6)*xi(2,
6) - xi(5,5)*xi(2,4)) + xi(6,6)*xi(2,4)\\


{5,6}|3\\ - (xi(2,6)*xi(1,5) - xi(2,5)*xi(1,6) - xi(4,5)*xi(3,5) + xi(4,6)*xi(3,
6) - xi(5,5)*xi(3,4)) + xi(6,6)*xi(3,4)\\


{5,6}|4\\xi(4,5)**2 + 1 - xi(4,6)**2 + xi(5,5)*xi(4,4) + xi(6,5)*xi(5,6) - (xi(5
,5) - xi(4,4))*xi(6,6)\\


{5,6}|5\\(xi(5,4) + xi(4,5))*xi(5,5) - xi(5,6)*xi(4,6) + xi(6,5)*xi(4,6) + (xi(5
,4) - xi(4,5))*xi(6,6)\\


{5,6}|6\\ - (xi(5,6)*xi(4,5) - xi(5,5)*xi(4,6) - xi(6,4)*xi(5,5) - xi(6,5)*xi(4,
5)) + (xi(6,4) - xi(4,6))*xi(6,6)\\


USD


\par Simultaneous resolution of the nonzero torsion equations and the matrix


equation USD J^2 = -I . USD


************************************************************************


Here we suppose that:


xi(1,1):=0


xi(1,2):=0


xi(1,3):=-1


xi(2,1):=0


xi(2,2):=xi(2,2)


xi(2,3):=0


xi(3,1):=1


xi(3,2):=0


xi(3,3):=0


xi(4,4):=0


xi(4,5):=0


xi(4,6):=-1


xi(5,4):=0


xi(5,5):= - xi(2,2)


xi(5,6):=0


xi(6,4):=1


xi(6,5):=0


xi(6,6):=0


************************************************************************


From the equation


%\\USD J^2(1,4):= - (xi(3,4) - xi(1,6))USD\\


\\ USD xi(3,4):=xi(1,6)USD


From the equation


%\\USD J^2(1,5):= - (xi(3,5) + xi(2,2)*xi(1,5))USD\\


\\ USD xi(3,5):= - xi(2,2)*xi(1,5)USD


From the equation


%\\USD J^2(1,6):= - (xi(3,6) + xi(1,4))USD\\


\\ USD xi(3,6):= - xi(1,4)USD


From the equation


%\\USD J^2(2,6):=xi(2,6)*xi(2,2) - xi(2,4)USD\\


\\ USD xi(2,4):=xi(2,6)*xi(2,2)USD


Now, From the equation


%{{{4,6},2},xi(1,6)**2 + xi(1,4)**2},


\\ USD xi(1,6):=0USD


\\ USD xi(1,4):=0USD


From the equation


%\\USD J^2(2,4):=(xi(2,2)**2 + 1)*xi(2,6)USD\\


\\ USD xi(2,6):=0USD


From the equation


%\\USD J^2(3,5):=(xi(2,2)**2 + 1)*xi(1,5)USD\\


\\ USD xi(1,5):=0USD


Now, From the equation


%\\USD J^2(4,1):= - (xi(6,1) - xi(4,3))USD\\


\\ USD xi(6,1):=xi(4,3)USD


From the equation


%\\USD J^2(4,2):= - (xi(6,2) - xi(4,2)*xi(2,2))USD\\


\\ USD xi(6,2):=xi(4,2)*xi(2,2)USD


From the equation


%\\USD J^2(4,3):= - (xi(6,3) + xi(4,1))USD\\


\\ USD xi(6,3):= - xi(4,1)USD


Now, From the equation


%{{{1,3},5},xi(4,3)**2 + xi(4,1)**2},


\\ USD xi(4,3):=0USD


\\ USD xi(4,1):=0USD


From the equation


%{{{2,5},4}, - (xi(2,2)**2 + 1)*xi(4,2)},


\\ USD xi(4,2):=0USD


Now, From the equation


%{{{1,2},5},xi(5,3)*xi(2,2) - xi(5,1)},


\\ USD xi(5,1):=xi(5,3)*xi(2,2)USD


Now, From the equation


%{{{2,3},5},(xi(2,2)**2 + 1)*xi(5,3)},


\\ USD xi(5,1):=0USD


Now, From the equation


%\\USD J^2(2,2):=xi(5,2)*xi(2,5) + xi(2,2)**2USD\\


\\ USD xi(5,2):=( - (xi(2,2)**2 + 1))/xi(2,5)USD


localrecap


USD \par Now the nonzero torsion equations left are :


{{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},0},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},0},
{{{1,3},1},0},
{{{1,3},2},0},
{{{1,3},3},0},
{{{1,3},4},0},
{{{1,3},5},0},
{{{1,3},6},0},
{{{1,4},1},0},
{{{1,4},2},0},
{{{1,4},3},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,4},6},0},
{{{1,5},1},0},
{{{1,5},2},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,5},5},0},
{{{1,5},6},0},
{{{1,6},1},0},
{{{1,6},2},0},
{{{1,6},3},0},
{{{1,6},4},0},
{{{1,6},5},0},
{{{1,6},6},0},
{{{2,3},1},0},
{{{2,3},2},0},
{{{2,3},3},0},
{{{2,3},4},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},1},0},
{{{2,4},2},0},
{{{2,4},3},0},
{{{2,4},4},0},
{{{2,4},5},0},
{{{2,4},6},0},
{{{2,5},1},0},
{{{2,5},2},0},
{{{2,5},3},0},
{{{2,5},4},0},
{{{2,5},5},0},
{{{2,5},6},0},
{{{2,6},1},0},
{{{2,6},2},0},
{{{2,6},3},0},
{{{2,6},4},0},
{{{2,6},5},0},
{{{2,6},6},0},
{{{3,4},1},0},
{{{3,4},2},0},
{{{3,4},3},0},
{{{3,4},4},0},
{{{3,4},5},0},
{{{3,4},6},0},
{{{3,5},1},0},
{{{3,5},2},0},
{{{3,5},3},0},
{{{3,5},4},0},
{{{3,5},5},0},
{{{3,5},6},0},
{{{3,6},1},0},
{{{3,6},2},0},
{{{3,6},3},0},
{{{3,6},4},0},
{{{3,6},5},0},
{{{3,6},6},0},
{{{4,5},1},0},
{{{4,5},2},0},
{{{4,5},3},0},
{{{4,5},4},0},
{{{4,5},5},0},
{{{4,5},6},0},
{{{4,6},1},0},
{{{4,6},2},0},
{{{4,6},3},0},
{{{4,6},4},0},
{{{4,6},5},0},
{{{4,6},6},0},
{{{5,6},1},0},
{{{5,6},2},0},
{{{5,6},3},0},
{{{5,6},4},0},
{{{5,6},5},0},
{{{5,6},6},0}}


\par The matrix USD J USD is :\\


USD  J^1_1=0;USD\\


USD  J^1_2=0;USD\\


USD  J^1_3=-1;USD\\


USD  J^1_4=0;USD\\


USD  J^1_5=0;USD\\


USD  J^1_6=0;USD\\


USD  J^2_1=0;USD\\


USD  J^2_2=xi(2,2);USD\\


USD  J^2_3=0;USD\\


USD  J^2_4=0;USD\\


USD  J^2_5=xi(2,5);USD\\


USD  J^2_6=0;USD\\


USD  J^3_1=1;USD\\


USD  J^3_2=0;USD\\


USD  J^3_3=0;USD\\


USD  J^3_4=0;USD\\


USD  J^3_5=0;USD\\


USD  J^3_6=0;USD\\


USD  J^4_1=0;USD\\


USD  J^4_2=0;USD\\


USD  J^4_3=0;USD\\


USD  J^4_4=0;USD\\


USD  J^4_5=0;USD\\


USD  J^4_6=-1;USD\\


USD  J^5_1=0;USD\\


USD  J^5_2=( - (xi(2,2)**2 + 1))/xi(2,5);USD\\


USD  J^5_3=0;USD\\


USD  J^5_4=0;USD\\


USD  J^5_5= - xi(2,2);USD\\


USD  J^5_6=0;USD\\


USD  J^6_1=0;USD\\


USD  J^6_2=0;USD\\


USD  J^6_3=0;USD\\


USD  J^6_4=1;USD\\


USD  J^6_5=0;USD\\


USD  J^6_6=0;USD\\


\\USD J^2(1,1):=-1USD\\


\\USD J^2(1,2):=0USD\\


\\USD J^2(1,3):=0USD\\


\\USD J^2(1,4):=0USD\\


\\USD J^2(1,5):=0USD\\


\\USD J^2(1,6):=0USD\\


\\USD J^2(2,1):=0USD\\


\\USD J^2(2,2):=-1USD\\


\\USD J^2(2,3):=0USD\\


\\USD J^2(2,4):=0USD\\


\\USD J^2(2,5):=0USD\\


\\USD J^2(2,6):=0USD\\


\\USD J^2(3,1):=0USD\\


\\USD J^2(3,2):=0USD\\


\\USD J^2(3,3):=-1USD\\


\\USD J^2(3,4):=0USD\\


\\USD J^2(3,5):=0USD\\


\\USD J^2(3,6):=0USD\\


\\USD J^2(4,1):=0USD\\


\\USD J^2(4,2):=0USD\\


\\USD J^2(4,3):=0USD\\


\\USD J^2(4,4):=-1USD\\


\\USD J^2(4,5):=0USD\\


\\USD J^2(4,6):=0USD\\


\\USD J^2(5,1):=0USD\\


\\USD J^2(5,2):=0USD\\


\\USD J^2(5,3):=0USD\\


\\USD J^2(5,4):=0USD\\


\\USD J^2(5,5):=-1USD\\


\\USD J^2(5,6):=0USD\\


\\USD J^2(6,1):=0USD\\


\\USD J^2(6,2):=0USD\\


\\USD J^2(6,3):=0USD\\


\\USD J^2(6,4):=0USD\\


\\USD J^2(6,5):=0USD\\


\\USD J^2(6,6):=-1USD\\


Trace(J):=0


J:=

[0           0           -1  0      0       0 ]
[                                             ]
[0        xi(2,2)        0   0   xi(2,5)    0 ]
[                                             ]
[1           0           0   0      0       0 ]
[                                             ]
[0           0           0   0      0       -1]
[                                             ]
[               2                             ]
[     - (xi(2,2)  + 1)                        ]
[0  -------------------  0   0   - xi(2,2)  0 ]
[         xi(2,5)                             ]
[                                             ]
[0           0           0   1      0       0 ]


J**2:=

[-1  0   0   0   0   0 ]
[                      ]
[0   -1  0   0   0   0 ]
[                      ]
[0   0   -1  0   0   0 ]
[                      ]
[0   0   0   -1  0   0 ]
[                      ]
[0   0   0   0   -1  0 ]
[                      ]
[0   0   0   0   0   -1]


det(J):=1


\\ Then USD J USD has entries :


USD  J^1_1=0;USD\\


USD  J^1_2=0;USD\\


USD  J^1_3=-1;USD\\


USD  J^1_4=0;USD\\


USD  J^1_5=0;USD\\


USD  J^1_6=0;USD\\


USD  J^2_1=0;USD\\


USD  J^2_2=xi(2,2);USD\\


USD  J^2_3=0;USD\\


USD  J^2_4=0;USD\\


USD  J^2_5=xi(2,5);USD\\


USD  J^2_6=0;USD\\


USD  J^3_1=1;USD\\


USD  J^3_2=0;USD\\


USD  J^3_3=0;USD\\


USD  J^3_4=0;USD\\


USD  J^3_5=0;USD\\


USD  J^3_6=0;USD\\


USD  J^4_1=0;USD\\


USD  J^4_2=0;USD\\


USD  J^4_3=0;USD\\


USD  J^4_4=0;USD\\


USD  J^4_5=0;USD\\


USD  J^4_6=-1;USD\\


USD  J^5_1=0;USD\\


USD  J^5_2=( - (xi(2,2)**2 + 1))/xi(2,5);USD\\


USD  J^5_3=0;USD\\


USD  J^5_4=0;USD\\


USD  J^5_5= - xi(2,2);USD\\


USD  J^5_6=0;USD\\


USD  J^6_1=0;USD\\


USD  J^6_2=0;USD\\


USD  J^6_3=0;USD\\


USD  J^6_4=1;USD\\


USD  J^6_5=0;USD\\


USD  J^6_6=0;USD\\


Hence we are finally reduced to 


\\


{\fontsize{8}{10} \selectfont


USDUSD J(\xi^2_2,\xi^2_5) := \begin{pmatrix}


0&


0&


-1&


0&


0&


0\\


0&


xi(2,2)&


0&


0&


xi(2,5)&


0\\


1&


0&


0&


0&


0&


0\\


0&


0&


0&


0&


0&


-1\\


0&


( - (xi(2,2)**2 + 1))/xi(2,5)&


0&


0&


 - xi(2,2)&


0\\


0&


0&


0&


1&


0&


0\end{pmatrix}USDUSD


}


with condition USD xi(2,5) \neq 0 USD.


\par Commutation relations of USD \mathfrak{m} : USD


USD[\tilde{x}_1,\tilde{x}_2]=xi(2,2)*tildex_1 + tildex_3USD;


USD[\tilde{x}_1,\tilde{x}_5]=xi(2,5)*tildex_1USD;


USD[\tilde{x}_2,\tilde{x}_3]= - (xi(2,2)*tildex_3 - tildex_1)USD;


USD[\tilde{x}_2,\tilde{x}_4]=((xi(2,2)**2 + 1)*tildex_4)/xi(2,5)USD;


USD[\tilde{x}_2,\tilde{x}_6]=((xi(2,2)**2 + 1)*tildex_6)/xi(2,5)USD;


USD[\tilde{x}_3,\tilde{x}_5]=xi(2,5)*tildex_3USD;


USD[\tilde{x}_4,\tilde{x}_5]= - (xi(2,2)*tildex_4 - tildex_6)USD;


USD[\tilde{x}_5,\tilde{x}_6]=xi(2,2)*tildex_6 + tildex_4USD;


\P


\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied


USD\forall x,y \in {\mathcal{G}}_{6,sl2bisxsl2bis},USD


\textit{i.e.} if USD{\mathcal{G}}_{6,sl2bisxsl2bis}USD


is a \textit{complex} algebra.


\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases{{{1,1}, - x(2)},
{{1,3},
((xi(2,2)**2 + 1)*x(5))/xi(2,5) - xi(2,2)*x(2)},
{{2,1}, - (x(3)*xi(2,2) + x(1))},
{{2,3}, - (x(3) - x(1)*xi(2,2))},
{{2,4},((xi(2,2)**2 + 1)*x(6))/xi(2,5)},
{{2,6}, - ((xi(2,2)**2 + 1)*x(4))/xi(2,5)},
{{3,1},
 - (x(5)*(xi(2,2)**2 + 1) - x(2)*xi(2,5)*xi(2,2))/xi(2,5)},
{{3,3}, - x(2)},
{{4,4}, - x(5)},
{{4,6},x(5)*xi(2,2) - x(2)*xi(2,5)},
{{5,1}, - xi(2,5)*x(3)},
{{5,3},xi(2,5)*x(1)},
{{5,4},x(6)*xi(2,2) - x(4)},
{{5,6}, - (x(6) + x(4)*xi(2,2))},
{{6,4}, - (x(5)*xi(2,2) - x(2)*xi(2,5))},
{{6,6}, - x(5)}}


\end{document}