%This is "program11.red"   (formerly "structcomplM14+1.tex") :
%computes the complex structures on the Lie algebra G_{M,14+1}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram11.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "6nilp/m.14+1"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
write "\setlength{\textwidth}{15cm}"$
write "\setlength{\textheight}{25cm}"$
write "\setlength{\voffset}{-2cm}" $
write " \setlength{\oddsidemargin}{0in}"$
write " \setlength{\evensidemargin}{0.36in}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
WRITE "This output from the file \texttt{program11.red}.\\"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
%write "\\ Case USD xi(4,3) =xi(2,1) USD."$
%WRITE "textbf{up to equivalence}"$
WRITE "\smallskip  \par "$
WRITE "Commutation relations for"$
write "USD {\mathcal{G}}_{", dim,",", PART(REFALGTEX,1), "}:USD\\"$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=X(i)*x(j)$
%FOR j:=1:DIM DO X(j):=MKID(x_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j) NEQ 0 THEN WRITE
 %       "USD[x_",i,",x_",j,"]=", X_(i,j),"USD;"$
        "USD[x(",i,"),x(",j,")]=", X_(i,j),"USD;"$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 operator x,f,A$
 noncom x,f$
 for all j let
  F(X(j))=FOR j1:=1:DIM SUM xi(j1,j)*X(j1)$
  let f(0)=0$
 for all j let f(-x(j))=-f(x(j))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(U*X(j))=U*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(U*V*X(j))=U*V*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(-U*V*X(j))=-U*V*f(X(j))$
 FOR ALL s,i,j,k,l
 LET f(X(s) *xi(i,j)*xi(k,l))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(xi(i,j)*xi(k,l)*X(s))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-X(s) *xi(i,j)*xi(k,l))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j))=(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j))=-(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j)**2)=(-xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j)**2)=(xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-xi(i,j)*xi(k,l)*X(s))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(-U*X(j))=-U*f(X(j))$
 FOR ALL A,B LET F(A+B)=F(A)+F(B)$
 %LIGNE A MODIFIER POUR LA LINEARITE SUR LE PARAMETRE CONTINU L
 for all j let f(L*x(j))=L*f(x(j))$
 for all j let f(-L*x(j))=-L*f(x(j))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%  collecting the torsion equations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for all j,k let
  temp110(j,k) = x(j)*F(x(k))$
 for all j,k let
  temp120(j,k) = F(temp110(j,k))$
 for all j,k let
  temp101(j,k) = F(x(j))*x(k)$
 for all j,k let
  temp102(j,k) = F(temp101(j,k))$
 for all j,k let
  temp2(j,k) =    F(x(j))* F(x(k)) $
 for all j,k let  temp0(j,k) =  x(j)*x(k)$
 for all j,k let
  A(j,k)=  -temp2(j,k) + temp0(j,k) + temp120(j,k) + temp102(j,k);

COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$

%IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion
%   in the following cases :",COLLECT_TORSION

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
ELSE WRITE "Zero torsion"$

write "\par"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$

%la liste des équations non nulles
%write "list of the nonzero torsion equations"$
COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_TORSION) JOIN
   FOR j2:=1:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_TORSION,j1),2) NEQ 0 THEN
   {{{PART(PART(COLLECT_TORSION,j1),1),j2}, V(j2)*PART(PART(COLLECT_TORSION,j1),2)} }
    ELSE {}$

%COMMENT
% WRITE "Torsion equations to cancel (Reduce output) : \\", COLLECT_EQ$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
  write "USD"$

%write "We don't write down again the torsion equations (see case 1)."$


 off factor$
 on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% computing the complex structures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "*****************************************************************"$
write "\\ \starline"$
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "First, one gets from  equation USD46|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", ws,"USD"$
write "\\ and from  equation USD56|2USD :"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\Then, if USD xi(3,6)=0 USD "$
xi(3,6):=0$
write "\\ USD xi(3,6):=", ws,"USD"$
write "\\ one gets from  equation USD26|5USD :"$
xi(5,6):=0$
write "\\ USD xi(5,6):=", ws,"USD"$
write "\\ and from  equation USD16|4USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", ws,"USD"$
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x6 entry in  USD J**2 USD  is USD xi(6,6)**2 USD"$
write "\\ hence, one gets a contradiction "$
write "\\ Hence USD xi(3,6)USD has to be USD \neq 0 .USD"$
clear  xi(3,6),xi(4,6),xi(5,6)$
write "\\ clear  USD xi(3,6),xi(4,6),xi(5,6) USD"$
write "\\ USD xi(3,6):=", xi(3,6), "USD"$
write "\\ USD xi(5,6):=", xi(5,6),"USD"$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then, one gets from  equation USD16|1USD :"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", ws,"USD"$
write "\\ and from  equation USD16|2USD :"$
xi(2,4):=0$
write "\\ USD xi(2,4):=", ws,"USD"$
write "\\ and from  equation USD26|1USD :"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", ws,"USD"$
write "\\ and from  equation USD26|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", ws,"USD"$
write "\\ and from  equation USD36|4USD :"$
xi(1,3):=0$
write "\\ USD xi(1,3):=", ws,"USD"$
write "\\ and from  equation USD36|5USD :"$
xi(2,3):=0$
write "\\ USD xi(2,3):=", ws,"USD"$
write "\\ and from  equation USD26|3USD :"$
xi(5,6):=-xi(3,5)$
write "\\ USD xi(5,6):=", ws,"USD"$
write "\\ and from  equation USD16|3USD :"$
xi(4,6):=-xi(3,4)$
write "\\ USD xi(4,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 1x1 entry in  USD J**2 USD  is "$
write "USD  {J**2}^1_1=xi(1,1)**2 + xi(2,1)*xi(1,2);USD\\"$
write "Hence USD xi(2,1)xi(1,2)\neq 0 USD"$
write "and USD xi(1,2) =(-1-xi(1,1)**2)/xi(2,1). USD"$
xi(1,2) :=(-1-xi(1,1)**2)/xi(2,1)$
write "\\ USD xi(1,2):=", ws,"USD"$
write "Moreover, the 1x2 entry in  USD J**2 USD  is "$
write "USD  (xi(2,2) + xi(1,1))*xi(1,2);USD hence\\"$
xi(2,2):= -xi(1,1)$
write "\\ USD xi(2,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\Suppose here  USD xi(1,1) \neq     0 USD "$
write "\\ Then, one gets from  equation USD14|4USD :"$
xi(3,4):=0$
write "\\ USD xi(3,4):=", ws,"USD"$
write "\\ and from  equation USD25|5USD :"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ Then, one gets from  equation USD13|3USD :"$
xi(4,3):=0$
write "\\ USD xi(4,3):=", ws,"USD"$
write "\\ and from  equation USD16|6USD :"$
xi(6,4):=0$
write "\\ USD xi(6,4):=", ws,"USD"$
write "\\ and from  equation USD23|3USD :"$
xi(5,3):=0$
write "\\ USD xi(5,3):=", ws,"USD"$
write "\\ and from  equation USD26|6USD :"$
xi(6,5):=0$
write "\\ USD xi(6,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD14|3USD :"$
write "USD xi(4,4)=-xi(1,1)"$
write "\\ and from  equation USD16|4USD :"$
write "USD xi(4,4)=xi(1,1)"$
write "\\ hence USD xi(1,1):=0. USD a contradiction"$
write "\\ Hence  one cannot have USD xi(1,1) \neq 0. USD "$
xi(1,1):=0$
write "\\ USD xi(1,1):=", ws,"USD"$
clear xi(3,4),xi(3,5),xi(5,3),xi(4,3),xi(6,4),xi(6,5)$
write "\\ Clear USDxi(3,4),xi(3,5),xi(5,3),xi(4,3),xi(6,4),xi(6,5) USD"$
write "\\ USD xi(3,4):=", xi(3,4),"USD"$
write "\\ USD xi(3,5):=", xi(3,5),"USD"$
write "\\ USD xi(4,3):=", xi(4,3),"USD"$
write "\\ USD xi(6,4):=", xi(6,4),"USD"$
write "\\ USD xi(5,3):=", xi(5,3),"USD"$
write "\\ USD xi(6,5):=", xi(6,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD14|3USD :"$
xi(4,4):=-xi(3,4)**2/xi(3,6)$
write "\\ USD xi(4,4):=", ws,"USD"$
write "\\ and from  equation USD25|3USD :"$
xi(5,5):=-xi(3,5)**2/xi(3,6)$
write "\\ USD xi(5,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD23|3USD :"$
%{2,3}|3\\(xi(3,5)*xi(3,3)*xi(2,1) - xi(3,4) + xi(5,3)*xi(3,6)*xi(2,1))/xi(2,1)\\
%$
 xi(3,4):= xi(3,5)*xi(3,3)*xi(2,1) + xi(5,3)*xi(3,6)*xi(2,1)$
write "\\ USD xi(3,4):=", ws,"USD"$
write "\\ and from  equation USD23|6USD :"$
%{2,3}|6\\( - (xi(6,4) - xi(4,3) - xi(6,5)*xi(3,3)*xi(2,1)) + xi(6,6)*xi(5,3)*xi(
%2,1))/xi(2,1)\\$
xi(4,3):=-( - xi(6,4) + xi(6,5)*xi(3,3)*xi(2,1) + xi(6,6)*xi(5,3)*xi(2,1))$
write "\\ USD xi(4,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD26|6USD :"$
%{2,6}|6\\ - (xi(5,3)*xi(3,6) + xi(3,5)*xi(3,3) - xi(6,5)*xi(3,6)) - xi(6,6)*xi(3
%,5)\\$
 xi(5,3):=( - xi(3,5)*xi(3,3) + xi(6,5)*xi(3,6) - xi(6,6)*xi(3,5))/xi(3,6)$
write "\\ USD xi(5,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD12|3USD :"$
%{1,2}|3\\xi(4,2)*xi(3,6) - xi(3,5)*xi(3,1) - xi(5,1)*xi(3,6) + xi(6,5)*xi(3,6)*
%xi(3,2)*xi(2,1) - xi(6,6)*xi(3,5)*xi(3,2)*xi(2,1)\\$
 xi(4,2):=-( - xi(3,5)*xi(3,1) - xi(5,1)*xi(3,6) + xi(6,5)*xi(3,6)*
xi(3,2)*xi(2,1) - xi(6,6)*xi(3,5)*xi(3,2)*xi(2,1))/xi(3,6)$
write "\\ USD xi(4,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD16|6USD :"$
%{1,6}|6\\(xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(6,6)*xi(2,1) + xi(6,4)*xi(3,6) +
% xi(3,5)*xi(2,1)\\$
%%{1,6}|6\\(xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(6,6)*xi(2,1) + xi(6,4)*xi(3,6) +
%% xi(3,5)*xi(2,1)\\$
 xi(6,4):=-((xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(6,6)*xi(2,1)
 %+ xi(6,4)*xi(3,6)
 + xi(3,5)*xi(2,1))/xi(3,6)$
write "\\ USD xi(6,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 3x1 entry in  USD J**2 USD  gives "$
%USD  {J**2}^3_1=xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(5,1)*xi(3,5) + xi(6,1)*xi
%(3,6) - (xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(4,1)*xi(2,1);USD\\$
%%USD  {J**2}^3_1=xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(5,1)*xi(3,5) + xi(6,1)*xi
%%(3,6) - (xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(4,1)*xi(2,1);USD\\$
xi(6,1):=-(xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(5,1)*xi(3,5)
%+ xi(6,1)*xi(3,6)
- (xi(6,6)*xi(3,5) - xi(6,5)*xi(3,6))*xi(4,1)*xi(2,1))/xi(3,6)$
write "\\ USD xi(6,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 3x6 entry in  USD J**2 USD  gives "$

%USD  {J**2}^3_6=xi(3,6)*xi(3,3) - xi(3,5)**2 + xi(6,6)*xi(3,6) - (xi(6,6)*xi(3,5
%) - xi(6,5)*xi(3,6))**2*xi(2,1)**2;USD\\$
%%USD  {J**2}^3_6=xi(3,6)*xi(3,3) + xi(3,5)**2 + xi(6,6)*xi(3,6) - (xi(6,6)*xi(3,5
%%) - xi(6,5)*xi(3,6))**2*xi(2,1)**2;USD\\$
xi(3,3):=-(-xi(3,5)**2 + xi(6,6)*xi(3,6) - (xi(6,6)*xi(3,5
) - xi(6,5)*xi(3,6))**2*xi(2,1)**2)/xi(3,6)$
write "\\ USD xi(3,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then, one gets from  equation USD15|3USD :"$
%{1,5}|3\\(xi(4,5) + xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5)
%**2*xi(2,1)\\$
%%{1,5}|3\\(xi(4,5) - xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5)
%%**2*xi(2,1)\\$
xi(4,5):= -((  xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5)
**2*xi(2,1))/xi(3,6)$
write "\\ USD xi(4,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then,   equation USD26|4USD reads :"$
write " USD {2,6}|4\\(-xi(3,6)*xi(2,1)**2 + xi(3,6))/xi(2,1)\\"$
write "Hence we get"$
xi(2,1)**2:=1$
write "\\ USD xi(2,1)**2:=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then,   equation USD12|6USD reads :"$
%{1,2}|6\\( - (xi(4,1)*xi(3,6) + xi(3,5)*xi(3,2) + xi(5,2)*xi(3,6) + xi(6,5)*xi(3
%,6)*xi(3,1)*xi(2,1)) + xi(6,6)*xi(3,5)*xi(3,1)*xi(2,1))/(xi(3,6)*xi(2,1))\\$
xi(5,2):=
 ( - xi(4,1)*xi(3,6) - xi(3,5)*xi(3,2)
 %- xi(5,2)*xi(3,6)
 - xi(6,5)*xi(3
,6)*xi(3,1)*xi(2,1) + xi(6,6)*xi(3,5)*xi(3,1)*xi(2,1))/xi(3,6)$
write "\\ USD xi(5,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then,   equation USD16|5USD gives :"$
%1,6}|5\\(xi(5,4) - xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5)
%*2*xi(2,1)\\$
xi(5,4):=-( (- xi(2,1) + xi(6,5)*xi(3,5)*xi(2,1))*xi(3,6) - xi(6,6)*xi(3,5)
**2*xi(2,1))/xi(3,6)$
write "\\ USD xi(5,4):=", ws,"USD"$


write "Then the 3x2 entry in  USD J**2 USD  gives "$
%USD  {J**2}^3_2=( - (xi(6,6)*xi(5,1)*xi(3,5) + xi(6,6)*xi(3,2)*xi(2,1) - xi(6,5)
%*xi(5,1)*xi(3,6) - xi(6,2)*xi(3,6)*xi(2,1) + xi(4,1)*xi(3,5)*xi(2,1) + xi(3,1)))
%/xi(2,1);USD\\$
xi(6,2):=
(xi(6,6)*xi(5,1)*xi(3,5) + xi(6,6)*xi(3,2)*xi(2,1) - xi(6,5)
*xi(5,1)*xi(3,6)
%- xi(6,2)*xi(3,6)*xi(2,1)
+ xi(4,1)*xi(3,5)*xi(2,1) + xi(3,1))
/(xi(3,6)*xi(2,1))$
write "\\ USD xi(6,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 3x3 entry in  USD J**2 USD  gives "$
%USD  {J**2}^3_3=( - (xi(6,6)**3*xi(3,5)**2 - 2*xi(6,6)**2*xi(6,5)*xi(3,6)*xi(3,5
%) - xi(6,6)**2*xi(3,6) + xi(6,6)*xi(6,5)**2*xi(3,6)**2 + xi(6,6)*xi(3,5)**2 - xi
%(6,3)*xi(3,6)**2))/xi(3,6);USD\\$
xi(6,3):=
(xi(6,6)**3*xi(3,5)**2 - 2*xi(6,6)**2*xi(6,5)*xi(3,6)*xi(3,5
) - xi(6,6)**2*xi(3,6) + xi(6,6)*xi(6,5)**2*xi(3,6)**2 + xi(6,6)*xi(3,5)**2
%- xi(6,3)*xi(3,6)**2
-xi(3,6)
)/xi(3,6)**2$
write "\\ USD xi(6,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "*****************************************************************"$
write "\\ \starline"$
write "At the present stage, we have : "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$

%WRITE "  ****Now the nonzero torsion equations left are :"$
WRITE " \\ \4stars Now the nonzero torsion equations left are :"$

COMMENT
write COLLECT_EQ$

%Latex output
 WRITE "\\ Torsion equations to cancel (Latex output) : USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>$
 write "USD"$
write "\\ \P \\"$
%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

 write "The matrix USD J USD is given by the following formula,"$
 write "with USD xi(1,2) = \pm 1   USD and "$
 write " USD xi(3,6) \neq 0  : USD"$
 write "\begin{equation} \label{M14+1general} \end{equation}"$
off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

%write "matJ:=",matj$

%on exp$
%off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


%  FOR i:=1:DIM DO FOR j:=1:DIM DO
%  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$
% WRITE "\\$ det J:=",DET(MATJ)$
% WRITE "Trace J:=",TRACE(MATJ)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
%write "{\fontsize{8}{10} \selectfont"$
 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$
%                         write "}"$
 %%%%%%%%%%%%%%%%%
 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\USD det J:=",DET(MATJ)," USD"$
 WRITE "USD Trace J:=",TRACE(MATJ)," USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "where \\"$
%write "USD a:=xi(3,1)", xi(3,1), ",\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%%
out "matJ11.red"$
WRITE "% This file contains the REDUCE data for the general complex structures"$
write "% on the Lie algebra g_{", dim, ",", PART(REFALGTEX,1), "}"$
WRITE "matrix J(6,6)"$
%WRITE "J:=",MATJ$
WRITE "% Writing the entries of J:"$
WRITE "% J(i,k) denotes the entry in the ith row , kth column"$
WRITE "% i.e. stands for  the  LaTex expression J^i_k"$
WRITE "% Similarly, xi(i,k) stands for the  LaTex expression \xi^i_k"$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "J(",i,",",j,"):=",MATJ(i,j)>>$


 %FOR i:=1:DIM DO FOR j:=1:DIM DO
 %<<WRITE MATJ(i,j):=MATJ(i,j)>>$

 WRITE "%where the parameters are subject to the following condition"$
 write "%USD xi(2,1)= \pm 1 , xi(3,6) \neq 0. USD"$
 write "xi(2,1)**2:=1"$

 write "END"$
 shut "matJ11.red"$
 out "rprogram11.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD "$

for each A in COLLECT_EQ  do if part(A,2) neq 0 then <<write part(A,1),"nonzero torsion">>$

for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$
if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off exp$
 on allfac$
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
%factor x(1),x(2),x(3),x(4),x(5),x(6)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM DO <<FACTOR MKID(tildex_,i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "\\ USD  [\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),";\\USD"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let BB(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM   JOIN    FOR j2:=1:DIM JOIN
   IF BB(j1,j2) NEQ 0 THEN
   {{{j1,j2},BB(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$
 write "\par Now we'll use equivalence by automorphisms."$
 write "All  automorphisms of"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "are  of the following  form  :"$

matrix phi(6,6)$
for i:=1:6 do for j:=1:6 do <<phi(i,j):=b(i,j)>>$

for i:=1:4 do for j:=5:6 do <<b(i,j):=0>>$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%b(1,1):=b(1,1)$

b(1,2):=b(2,1)*u$

b(1,3):=0$

b(1,4):=0$

b(1,5):=0$

b(1,6):=0$

%b(2,1):=b(2,1)$

b(2,2):= - b(1,1)*u$

b(2,3):=0$

b(2,4):=0$

b(2,5):=0$

b(2,6):=0$

b(3,1):=0$

b(3,2):=0$

%b(3,3):=b(3,3)$

b(3,4):=0$

b(3,5):=0$

b(3,6):=0$

%b(4,1):=b(4,1)$

%b(4,2):=b(4,2)$

%b(4,3):=b(4,3)$

b(4,4):=b(3,3)*b(1,1)$

b(4,5):=b(3,3)*b(2,1)*u$

b(4,6):=0$

b(5,1):=u*( - b(4,2) + b(2,1)*k)$

b(5,2):=b(4,1)*u - b(1,1)*k$

%b(5,3):=b(5,3)$

b(5,4):=b(3,3)*b(2,1)$

b(5,5):= - b(3,3)*b(1,1)*u$

b(5,6):=0$

%b(6,1):=b(6,1)$

%b(6,2):=b(6,2)$

%b(6,3):=b(6,3)$

b(6,4):=b(5,3)*b(2,1) + b(4,3)*b(1,1)$

b(6,5):=u*( - b(5,3)*b(1,1) + b(4,3)*b(2,1))$

b(6,6):=b(3,3)*(b(2,1)**2 + b(1,1)**2)$

u**2:=1$
%\\$ det phi:=(b(2,1)**2 + b(1,1)**2)**3*b(3,3)**4$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "where USD \det \Phi:=(b(2,1)**2 + b(1,1)**2)**3*b(3,3)**4 \neq 0 USD"$
write "and USD u= \pm 1, \; k \in \Rmath."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on exp$
off factor$

write "\textit{Reduce} here checks that USD \Phi USD is indeed an automorphism:\\"$

for all i let phi_x(i) = for j:=1:6 sum phi(j,i)*x(j)$

WS:={
{{1,2},phi_x(1)*phi_x(2)},
{{1,3},phi_x(1)*phi_x(3)-phi_x(4)},
{{1,4},phi_x(1)*phi_x(4)-phi_x(6)},
{{1,5},phi_x(1)*phi_x(5)},
{{1,6},phi_x(1)*phi_x(6)},
{{2,3},phi_x(2)*phi_x(3)-phi_x(5)},
{{2,4},phi_x(2)*phi_x(4)},
{{2,5},phi_x(2)*phi_x(5)-phi_x(6)},
{{2,6},phi_x(2)*phi_x(6)},
{{3,4},phi_x(3)*phi_x(4)},
{{3,5},phi_x(3)*phi_x(5)},
{{3,6},phi_x(3)*phi_x(6)},
{{4,5},phi_x(4)*phi_x(5)},
{{4,6},phi_x(4)*phi_x(6)},
{{5,6},phi_x(5)*phi_x(6)}} $

COLLECT_aut:=FOR each A in ws JOIN
   IF PART(A,2) NEQ 0 THEN {PART(A,1),PART(A,2)}
   ELSE {}$


IF LENGTH(COLLECT_aut) NEQ 0 THEN <<WRITE "USD \Phi USD is not an automorphism
 in the following cases",COLLECT_aut>>
 else <<WRITE " USD \surd \Phi USD is  an automorphism">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has the following selected entries :"$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Take the following values :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
u:=-1$
write "\\ USD u:=", ws,"USD"$
b(1,1):=1$
write "\\ USD b(1,1):=", ws,"USD"$
b(2,1):=0$
write "\\ USD b(2,1):=", ws,"USD"$
b(3,3):=1$
write "\\ USD b(3,3):=", ws,"USD"$

%\\USD J2(3,1):=(b(6,1)*xi(3,6) - b(4,2)*xi(3,5)*u - b(4,1)*xi(6,6)*xi(3,5)*xi(2,
%1) + b(4,1)*xi(6,5)*xi(3,6)*xi(2,1) + b(2,1)*xi(3,5)*k*u + b(2,1)*xi(3,2) + b(1,
%1)*xi(3,1))/b(3,3)USD\\$

b(6,1):=-(
%b(6,1)*xi(3,6)
- b(4,2)*xi(3,5)*u - b(4,1)*xi(6,6)*xi(3,5)*xi(2,
1) + b(4,1)*xi(6,5)*xi(3,6)*xi(2,1) + b(2,1)*xi(3,5)*k*u + b(2,1)*xi(3,2) + b(1,
1)*xi(3,1))/xi(3,6)$
write "\\ USD b(6,1):=", ws,"USD"$

%\\USD J2(3,2):=(b(6,2)*xi(3,6) - b(4,2)*xi(6,6)*xi(3,5)*xi(2,1) + b(4,2)*xi(6,5)
%*xi(3,6)*xi(2,1) + b(4,1)*xi(3,5)*u + b(2,1)*xi(3,1)*u - b(1,1)*xi(3,5)*k - b(1,
%1)*xi(3,2)*u)/b(3,3)USD\\$

b(6,2):=
-(
%b(6,2)*xi(3,6)
- b(4,2)*xi(6,6)*xi(3,5)*xi(2,1) + b(4,2)*xi(6,5)
*xi(3,6)*xi(2,1) + b(4,1)*xi(3,5)*u + b(2,1)*xi(3,1)*u - b(1,1)*xi(3,5)*k - b(1,
1)*xi(3,2)*u)/xi(3,6)$
write "\\ USD b(6,2):=", ws,"USD"$

%\\USD J2(3,5):=(u*( - b(5,3)*b(1,1)*xi(3,6) + b(4,3)*b(2,1)*xi(3,6) - b(3,3)*b(2
%,1)*xi(6,6)*xi(3,5)*xi(2,1) + b(3,3)*b(2,1)*xi(6,5)*xi(3,6)*xi(2,1) - b(3,3)*b(1
%,1)*xi(3,5)))/b(3,3)USD\\$

b(5,3):=
(
%- b(5,3)*b(1,1)*xi(3,6)
+ b(4,3)*b(2,1)*xi(3,6) - b(3,3)*b(2
,1)*xi(6,6)*xi(3,5)*xi(2,1) + b(3,3)*b(2,1)*xi(6,5)*xi(3,6)*xi(2,1) - b(3,3)*b(1
,1)*xi(3,5))/(xi(3,6)*b(1,1))$
write "\\ USD b(5,3):=", ws,"USD"$

%\\USD J2(6,6):=( - b(6,3)*b(3,3)*xi(3,6) + b(5,3)**2*xi(3,6) + b(5,3)*b(3,3)*xi(
%3,5) + b(4,3)**2*xi(3,6) - b(4,3)*b(3,3)*xi(6,6)*xi(3,5)*xi(2,1) + b(4,3)*b(3,3)
%*xi(6,5)*xi(3,6)*xi(2,1) + b(3,3)**2*xi(6,6))/b(3,3)**2USD\\$

b(6,3):=
(
%- b(6,3)*b(3,3)*xi(3,6)
+ b(5,3)**2*xi(3,6) + b(5,3)*b(3,3)*xi(
3,5) + b(4,3)**2*xi(3,6) - b(4,3)*b(3,3)*xi(6,6)*xi(3,5)*xi(2,1) + b(4,3)*b(3,3)
*xi(6,5)*xi(3,6)*xi(2,1) + b(3,3)**2*xi(6,6))/(b(3,3)*xi(3,6))$
write "\\ USD b(6,3):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\\USD J2(4,1):=( - 2*b(4,2)*xi(3,6)*xi(2,1) - xi(6,6)*xi(3,5)*xi(3,1)*xi(2,1) +
%xi(6,5)*xi(3,6)*xi(3,1)*xi(2,1) + xi(4,1)*xi(3,6))/xi(3,6)USD\\$
b(4,2):=
(
%- 2*b(4,2)*xi(3,6)*xi(2,1)
- xi(6,6)*xi(3,5)*xi(3,1)*xi(2,1) +
xi(6,5)*xi(3,6)*xi(3,1)*xi(2,1) + xi(4,1)*xi(3,6))/(2*xi(3,6)*xi(2,1))$
write "\\ USD b(4,2):=", ws,"USD"$


%\\USD J2(5,1):=(2*b(4,1)*xi(3,6)*xi(2,1) + xi(5,1)*xi(3,6) + xi(3,6)*xi(2,1)*k +
% xi(3,5)*xi(3,1))/xi(3,6)USD\\$

b(4,1):= -
(
%2*b(4,1)*xi(3,6)*xi(2,1)
+ xi(5,1)*xi(3,6) + xi(3,6)*xi(2,1)*k +
 xi(3,5)*xi(3,1))/(2*xi(3,6)*xi(2,1))$
write "\\ USD b(4,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$


%write "\\"$
% write "USDUSD \Phi = \begin{pmatrix}"$
% for i:=1:dim do
%     if i neq dim then
%         for j:=1:dim do
%                         if j neq dim then <<write phi(i,j), "&">>
%                         else    <<write phi(i,j), "\\">>
%     else
%         for j:=1:dim do
%                         if j neq dim then <<write phi(i,j), "&">>
%                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%

% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  \Phi^",i,"_",j,"=",phi(i,j),";USD\\">>$

%write "phi:=",phi$

off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Hence, we are led to the case where"$
xi(3,1):=0$
write "\\ USD xi(3,1):=", ws,"USD"$
xi(3,2):=0$
write "\\ USD xi(3,2):=", ws,"USD"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", ws,"USD"$
xi(4,1):=0$
write "\\ USD xi(4,1):=", ws,"USD"$
xi(5,1):=0$
write "\\ USD xi(5,1):=", ws,"USD"$
xi(6,6):=0$
write "\\ USD xi(6,6):=", ws,"USD"$
write "clear USD u, b(1,1),b(2,1),b(3,3),b(6,1),b(6,2),b(5,3),b(6,3),b(4,1),b(4,2) USD"$
clear  u, b(1,1),b(2,1),b(3,3),b(6,1),b(6,2),b(5,3),b(6,3),b(4,1),b(4,2) $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
%write "{\fontsize{8}{10} \selectfont"$
 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$
%                         write "}"$
 %%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Take now the following values :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
u:=-xi(2,1)$
write "\\ USD u:=", ws,"USD"$
b(1,1)**2:=1/(abs(xi(3,6)))$
b(2,1):=0$
b(3,3):=1$
b(4,3):=-xi(6,5)*xi(2,1)$
b(5,3):=0$

%\\USD J2(3,1):=xi(3,6)*(b(6,1) + b(4,1)*xi(6,5)*xi(2,1))USD\\$
b(6,1) :=- b(4,1)*xi(6,5)*xi(2,1)$

%\\USD J2(3,2):=xi(3,6)*(b(6,2) + b(4,2)*xi(6,5)*xi(2,1))USD\\$
b(6,2):=- b(4,2)*xi(6,5)*xi(2,1)$

%\\USD J2(4,1):=( - 2*b(4,2))/b(1,1)USD\\$
b(4,2):=0$


%\\USD J2(5,1):=(2*b(4,1) + b(1,1)*xi(2,1)*k)/b(1,1)USD\\$
b(4,1):=-  b(1,1)*xi(2,1)*k/2$

%\\USD J2(6,6):= - b(6,3)*xi(3,6)USD\\$
b(6,3):=0$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$


r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Hence, we are led to the case where moreover "$
xi(6,5):=0$
write "\\ USD xi(6,5):=", ws,"USD"$
xi(2,1):=1$
write "\\ USD xi(2,1):=", ws,"USD"$
xi(3,6)**2:=1$
write "\\ USD xi(3,6)**2:=", ws,"USD"$
write "clear USD u, b(1,1)**2,b(2,1),b(3,3),b(4,3),b(5,3),b(6,1),b(6,2),b(6,3),b(4,1),b(4,2) USD"$
clear  u, b(1,1)**2,b(2,1),b(3,3),b(4,3),b(5,3),b(6,1),b(6,2),b(6,3),b(4,1),b(4,2) $
write "Then we have"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
%write "{\fontsize{8}{10} \selectfont"$
 write "USDUSD J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$
%                         write "}"$
 %%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$


r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence any   USD J USD in \ref{M14+1general} "$
write "is equivalent to  "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
write "\begin{equation}"$
write "\label{M14+1final}"$
% write "USDUSD J = \begin{pmatrix}"$
 write " J(xi(3,6)) = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}" >>$
 write "\end{equation}"$
 write "}"$
write " where USD xi(3,6)= \pm 1.\\USD"$
write "The 2 matrices corresponding to USD xi(3,6)= \pm 1\\USD"$
write "are not equivalent."$
 %%%%%%%%%%%%%%%%%
 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\USD det J:=",DET(MATJ)," USD"$
 WRITE "USD Trace J:=",TRACE(MATJ)," USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD "$

for each A in COLLECT_EQ  do if part(A,2) neq 0 then <<write part(A,1),"nonzero torsion">>$

for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$
if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% off exp$
% on allfac$
on exp$
off factor$
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
%factor x(1),x(2),x(3),x(4),x(5),x(6)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM DO <<FACTOR MKID(tildex_,i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "\\ USD  [\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),";\\USD"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let BB(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM   JOIN    FOR j2:=1:DIM JOIN
   IF BB(j1,j2) NEQ 0 THEN
   {{{j1,j2},BB(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \starline"$
write "\end{document}"$
bye$