%This is "program5.red"   (formerly "structcompl6_4tex.tex") :
%computes the complex structures on the Lie algebra G_{6,4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram5.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%   ENTRER LE FICHIER DE L'ALGEBRE %%%%%%%%%%%%%%%%%%%%%
%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "6nilp/6nilp.4"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
WRITE "This output from the file \texttt{structcompl6\_4tex.tex}"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
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"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% computing the complex structures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
write "\\ One first gets"$
write "\\ from equation USD36|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", xi(1,6),"USD"$
write "\\ and from equation USD45|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", xi(2,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write " If USD xi(2,6) \neq 0, USD"$
write "then, one gets from  equation USD56|5USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
write "\\ and from equation USD56|4USD :"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", xi(1,5),"USD"$
write "\\ and from equation USD46|3USD :"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", xi(3,5),"USD"$
write "\\ and from equation USD36|4USD :"$
xi(1,3):=0$
write "\\ USD xi(1,3):=", xi(1,3),"USD"$
write "\\ and from equation USD34|2USD :"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", xi(1,4),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD16|2USD :"$
xi(3,6):=-xi(2,4)$
write "\\ USD xi(3,6):=", xi(3,6),"USD"$
write "\\ and from equation USD23|2USD :"$
xi(1,2):=0$
write "\\ USD xi(1,2):=", xi(1,2),"USD"$
write "\\ and from equation USD46|6USD :"$
xi(6,5):=0$
write "\\ USD xi(6,5):=", xi(6,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then the 1x1 entry in  USD J**2 USD  is USD xi(1,1)**2 USD,"$
write "and the 5x5 entry in  USD J**2 USD  is USD xi(5,5)**2 USD,"$
write "which doesn't make sense."$
write "Hence USD xi(2,6)=0 USD\\"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", xi(2,6),"USD\\"$
clear xi(4,5),xi(1,5),xi(3,5),xi(1,3),xi(1,4),xi(3,6),xi(1,2),xi(6,5)$
write "we clear USDxi(4,5),xi(1,5),xi(3,5),xi(1,3),xi(1,4),xi(3,6),xi(1,2),xi(6,5): USD"$
write "\\ USD xi(4,5):=", xi(4,5),"USD\\"$
write "\\ USD xi(1,5):=", xi(1,5),"USD\\"$
write "\\ USD xi(3,5):=", xi(3,5),"USD\\"$
write "\\ USD xi(1,3):=", xi(1,3),"USD\\"$
write "\\ USD xi(1,4):=", xi(1,4),"USD\\"$
write "\\ USD xi(3,6):=", xi(3,6),"USD\\"$
write "\\ USD xi(1,2):=", xi(1,2),"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|2USD :"$
xi(2,4):=0$
write "\\ USD xi(2,4):=", xi(2,4),"USD"$
write "\\ and from equation USD16|3USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", xi(3,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write " If USD xi(2,3) \neq 0, USD"$
write "then, one gets from  equation USD13|1USD :"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", xi(1,4),"USD"$
write "\\ and from equation USD13|3USD :"$
xi(3,4):=0$
write "\\ USD xi(3,4):=", xi(3,4),"USD"$
write "\\ and from equation USD34|3USD :"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", xi(3,5),"USD"$
write "\\ and from equation USD36|5USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " If moreover USD xi(1,5) \neq 0, USD"$
write "then, one gets from  equation USD12|1USD :"$
xi(4,1):=0$
write "\\ USD xi(4,1):=", xi(4,1),"USD"$
write "\\ and from equation USD14|1USD :"$
xi(2,1):=0$
write "\\ USD xi(2,1):=", xi(2,1),"USD"$
write "\\ and from equation USD15|6USD :"$
xi(3,1):=0$
write "\\ USD xi(3,1):=", xi(3,1),"USD"$
write "\\ and from equation USD23|3USD :"$
xi(4,3):=0$
write "\\ USD xi(4,3):=", xi(4,3),"USD"$
write "\\ and from equation USD24|1USD :"$
xi(4,4):=-xi(2,2)$
write "\\ USD xi(4,4):=", xi(4,4),"USD"$
write "\\ and from equation USD25|1USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
write "\\ and from equation USD34|1USD :"$
xi(2,3):=0$
write "\\ USD xi(2,3):=", xi(2,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then equation USD24|5USD reads : USD xi(2,2)**2 +1 = 0USD"$
write "which doesn't make sense."$
write "Hence USD xi(1,5)=0 USD\\"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", xi(1,5),"USD\\"$
clear xi(4,1),xi(2,1),xi(3,1),xi(4,3),xi(4,4),xi(4,5),xi(2,3)$
write "we clear USDxi(4,1),xi(2,1),xi(3,1),xi(4,3),xi(4,4),xi(4,5),xi(2,3): USD"$
write "\\ USD xi(4,1):=", xi(4,1),"USD\\"$
write "\\ USD xi(2,1):=", xi(2,1),"USD\\"$
write "\\ USD xi(3,1):=", xi(3,1),"USD\\"$
write "\\ USD xi(4,3):=", xi(4,3),"USD\\"$
write "\\ USD xi(4,4):=", xi(4,4),"USD\\"$
write "\\ USD xi(4,5):=", xi(4,5),"USD\\"$
write "\\ USD xi(2,3):=", xi(2,3),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD25|4USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then the 4x4 entry in  USD J**2 USD  is USD xi(4,4)**2 USD,"$
write "which doesn't make sense."$
write "Hence we've got that USD xi(2,3)=0 USD\\"$
xi(2,3):=0$
write "\\ USD xi(2,3):=", xi(2,3),"USD\\"$
clear xi(4,5),xi(1,4),xi(3,4),xi(3,5),xi(4,6)$
write "we clear USDxi(4,5),xi(1,4),xi(3,4),xi(3,5),xi(4,6): USD"$
write "\\ USD xi(4,5):=", xi(4,5),"USD\\"$
write "\\ USD xi(1,4):=", xi(1,4),"USD\\"$
write "\\ USD xi(3,4):=", xi(3,4),"USD\\"$
write "\\ USD xi(3,5):=", xi(3,5),"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 USD24|1USD :"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", xi(1,4),"USD"$
write "\\ and from equation USD25|4USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write " If USD xi(3,5) \neq 0, USD"$
write "then, one gets from  equation USD14|3USD :"$
xi(2,1):=0$
write "\\ USD xi(2,1):=", xi(2,1),"USD"$
write "\\ and from equation USD15|4USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
write "\\ and from equation USD15|5USD :"$
xi(5,6):=0$
write "\\ USD xi(5,6):=", xi(5,6),"USD"$
write "\\ and from equation USD15|6USD :"$
xi(6,6):=xi(1,1)$
write "\\ USD xi(6,6):=", xi(6,6),"USD"$
write "\\ and from equation USD24|3USD :"$
xi(4,4):=-xi(2,2)$
write "\\ USD xi(4,4):=", xi(4,4),"USD"$
write "\\ and from equation USD25|6USD :"$
xi(1,2):=0$
write "\\ USD xi(1,2):=", xi(1,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then equation USD24|5USD reads : USD xi(2,2)**2 +1 = 0USD"$
write "which doesn't make sense."$
write "Hence USD xi(3,5)=0 USD\\"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", xi(3,5),"USD\\"$
clear xi(2,1),xi(4,6),xi(5,6),xi(6,6),xi(4,4),xi(1,2)$
write "we clear USDxi(2,1),xi(4,6),xi(5,6),xi(6,6),xi(4,4),xi(1,2) : USD"$
write "\\ USD xi(2,1):=", xi(2,1),"USD\\"$
write "\\ USD xi(4,6):=", xi(4,6),"USD\\"$
write "\\ USD xi(5,6):=", xi(5,6),"USD\\"$
write "\\ USD xi(6,6):=", xi(6,6),"USD\\"$
write "\\ USD xi(4,4):=", xi(4,4),"USD\\"$
write "\\ USD xi(1,2):=", xi(1,2),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write " If USD xi(4,6) \neq 0, USD"$
write "then, one gets from  equation USD14|4USD :"$
xi(3,4):=0$
write "\\ USD xi(3,4):=", xi(3,4),"USD"$
write "\\ and from equation USD16|5USD :"$
xi(2,1):=0$
write "\\ USD xi(2,1):=", xi(2,1),"USD"$
write "\\ and from equation USD26|6USD :"$
xi(6,5):=0$
write "\\ USD xi(6,5):=", xi(6,5),"USD"$
write "\\ and from equation USD26|5USD :"$
xi(5,5):=xi(2,2)$
write "\\ USD xi(5,5):=", xi(5,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then equation USD24|5USD reads : USD xi(2,2)**2 +1 = 0USD"$
write "which doesn't make sense."$
write "Hence USD xi(4,6)=0 USD\\"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD\\"$
clear xi(3,4),xi(2,1),xi(6,5),xi(5,5)$
write "we clear USDxi(3,4),xi(2,1),xi(6,5),xi(5,5) : USD"$
write "\\ USD xi(3,4):=", xi(3,4),"USD\\"$
write "\\ USD xi(2,1):=", xi(2,1),"USD\\"$
write "\\ USD xi(6,5):=", xi(6,5),"USD\\"$
write "\\ USD xi(5,5):=", xi(5,5),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 2x3 entry in  USD J**2 USD  is USD xi(2,1) xi(1,3) USD,"$
write "and the 2x2 entry in  USD J**2 USD  is USD xi(2,2)**2 +xi(2,1)xi(1,2) USD."$
write "Hence if USD xi(1,3) \neq 0,USD then USD xi(2,1) =0 USD"$
write "and the 2x2 entry cannot be equal to USD -1USD\\"$
write "Hence we've got that USD xi(1,3)=0 USD\\"$
xi(1,3):=0$
write "\\ USD xi(1,3):=", xi(1,3),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "USD \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "The USD 1  \times 1 USD term in  USD J^2 USD is :\\ "$
write "USD  {J**2}^1_1=xi(2,1)*xi(1,2) + xi(1,1)**2;USD\\"$
write "The USD 3  \times 3 USD term in  USD J^2 USD is : \\"$
write "USD  {J**2}^3_3=xi(4,3)*xi(3,4) + xi(3,3)**2;USD\\"$
write "The USD 5 \times 5 USD term in  USD J^2 USD is :\\ "$
write "USD  {J**2}^5_5=xi(6,5)*xi(5,6) + xi(5,5)**2.USD\\"$
write "Hence USD xi(1,2)*xi(2,1),xi(3,4)*xi(4,3), xi(5,6)*xi(6,5) USD are USD"$
WRITE "\neq 0 USD.\\"$
write "From the USD 1  \times 2 USD term in  USD J^2 USD one gets : "$
xi(2,2):=-xi(1,1)$
write "\\ USD xi(2,2):=", xi(2,2),"USD"$
write "From the USD 3  \times 4 USD term in  USD J^2 USD one gets : "$
xi(4,4):=-xi(3,3)$
write "\\ USD xi(4,4):=", xi(3,3),"USD"$
write "From the USD 5  \times 6 USD term in  USD J^2 USD one gets : "$
xi(6,6):=-xi(5,5)$
write "\\ USD xi(6,6):=", xi(6,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "From the diagonal terms of USD J**2 USD one gets :"$
%termes diagonaux de J**2  imposent
xi(2,1):=-(xi(1,1)**2 +1)/xi(1,2)$
%write "xi(2,1):=",ws$
write "\\ USD xi(2,1):=", xi(2,1),"USD"$
xi(4,3):=-(xi(3,3)**2 +1)/xi(3,4)$
%write "xi(4,3):=",ws$
write "\\ USD xi(4,3):=", xi(4,3),"USD"$
xi(6,5):=-(xi(5,5)**2 +1)/xi(5,6)$
%write "xi(6,5):=",ws$
write "\\ USD xi(6,5):=", xi(6,5),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then, one gets from  equation USD12|6USD :"$
  xi(4,1):=-  ( - xi(5,6)*xi(5,5)*xi(3,2) - xi(5,6)*xi(3,2)*xi(1,1)
    + xi(5,6)*xi(3,1)*xi(1,2))/ (xi(5,5)**2+1)$
write "\\ USD xi(4,1):=", xi(4,1),"USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "USD \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 3  \times 2 USD term in  USD J^2 USD one gets : "$
%USD  J**2^3_2=xi(4,2)*xi(3,4) + xi(3,3)*xi(3,2) - xi(3,2)*xi(1,1) + xi(3,1)*xi(1
%,2);USD\\$
xi(3,1):= -(xi(4,2)*xi(3,4) + xi(3,3)*xi(3,2) - xi(3,2)*xi(1,1))/xi(1,2)$
write "\\ USD xi(3,1):=", xi(3,1),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "USD \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5  \times 2 USD term in  USD J^2 USD one gets : "$
%USD  J**2^5_2=xi(6,2)*xi(5,6) + xi(5,5)*xi(5,2) + xi(5,4)*xi(4,2) + xi(5,3)*xi(3
%,2) - xi(5,2)*xi(1,1) + xi(5,1)*xi(1,2);USD\\$
 xi(5,1):=-(
xi(6,2)*xi(5,6) + xi(5,5)*xi(5,2) + xi(5,4)*xi(4,2) + xi(5,3)*xi(3,2) - xi(5,2)*xi(1,1))
/xi(1,2)$
write "\\ USD xi(5,1):=", xi(5,1),"USD"$
write "And from the USD 5  \times 4 USD term in  USD J^2 USD one gets : "$
%USD  J**2^5_4=xi(6,4)*xi(5,6) + xi(5,5)*xi(5,4) - xi(5,4)*xi(3,3) + xi(5,3)*xi(3,4);USD\\$
 xi(5,3):=-(
xi(6,4)*xi(5,6) + xi(5,5)*xi(5,4) - xi(5,4)*xi(3,3))/xi(3,4)$
write "\\ USD xi(5,3):=", xi(5,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "USD \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 5  \times 3 USD term in  USD J^2 USD one gets : "$
%USD  J**2^5_3=( - xi(6,4)*xi(5,6)*xi(5,5) - xi(6,4)*xi(5,6)*xi(3,3) + xi(6,3)*xi
%(5,6)*xi(3,4) - xi(5,5)**2*xi(5,4) - xi(5,4))/xi(3,4);USD\\$
 xi(5,4):=
( - xi(6,4)*xi(5,6)*xi(5,5) - xi(6,4)*xi(5,6)*xi(3,3) + xi(6,3)*xi(5,6)*xi(3,4) )
/(xi(5,5)**2+1)$
write "\\ USD xi(5,4):=", xi(5,4),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "USD \par \textit{Reduce} now computes the matrix  USD J^2 USD,"$
write "which must be equal to USD -I USD."$
write "From the USD 6  \times 2 USD term in  USD J^2 USD one gets : "$
%USD  J**2^6_2=(xi(6,4)*xi(5,6)*xi(4,2) + xi(6,3)*xi(5,6)*xi(3,2) - xi(6,2)*xi(5,
%6)*xi(5,5) - xi(6,2)*xi(5,6)*xi(1,1) + xi(6,1)*xi(5,6)*xi(1,2) - xi(5,5)**2*xi(5
%,2) - xi(5,2))/xi(5,6);USD\\$
xi(5,2):=
(xi(6,4)*xi(5,6)*xi(4,2) + xi(6,3)*xi(5,6)*xi(3,2) - xi(6,2)*xi(5,
6)*xi(5,5) - xi(6,2)*xi(5,6)*xi(1,1) + xi(6,1)*xi(5,6)*xi(1,2)
)/(xi(5,5)**2+1)$
write "\\ USD xi(5,2):=", xi(5,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then, one gets from  equation USD14|5USD :"$
%{{{1,4},5},
%(xi(5,6)*xi(3,4)*xi(1,2) - xi(5,5)*xi(1,1)**2 - xi(5,5) - xi(3,3)*xi(1,1)**2 -
%xi(3,3))/xi(1,2)},
xi(3,3):=
(xi(5,6)*xi(3,4)*xi(1,2) - xi(5,5)*xi(1,1)**2 - xi(5,5))/(xi(1,1)**2+1)$
write "\\ USD xi(3,3):=", xi(3,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write " If USD xi(5,5)=-xi(1,1) , USD"$
write "then, equation USD13|5USD reads USDxi(1,1)**2 + 1 = 0USD:"$
write "which doesn't make sense."$
write "Hence USD xi(5,5)\neq -xi(1,1) USD\\"$
write "Then, equation USD13|5USD gives:"$
%{{{1,3},5},
%(xi(5,6)*xi(5,5)*xi(3,4)*xi(1,2) + xi(5,6)*xi(3,4)*xi(1,2)*xi(1,1) - xi(5,5)**2*
%xi(1,1)**2 - xi(5,5)**2 - xi(1,1)**2 - 1)/(xi(3,4)*xi(1,2))},
xi(5,6):=-(- xi(5,5)**2* xi(1,1)**2 - xi(5,5)**2 - xi(1,1)**2 - 1)/(xi(3,4)*xi(1,2)*(xi(5,5)+xi(1,1)))$
write "\\ USD xi(5,6):=", xi(5,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

WRITE "USD \par 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$

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

 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 "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 "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$

%%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%%
out "matJ5.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(1,2)*xi(3,4)*(xi(1,1) + xi(5,5)) \neq 0. USD"$

 write "END"$
 shut "matJ5.red"$
 out "rprogram5.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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 form"$

matrix phi(6,6)$
for i:=1:3 do for j:=1:i do <<phi(i,j):=b(i,j)>>$

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

b(1,2):=0$
b(1,3):=0$
b(2,1):=0$
b(2,3):=0$
b(4,3):=0$
b(4,4):=b(1,1)*b(2,2)$
b(4,5):=0$
b(4,6):=0$
b(5,4):=-b(4,1)*b(2,2)$
b(5,5):=b(1,1)*b(2,2)**2$
b(5,6):=0$
b(6,4):=b(1,1)*b(3,2)$
b(6,5):=0$
b(6,6):=b(1,1)*b(3,3)$

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 "\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)-phi_x(4)},
{{1,3},phi_x(1)*phi_x(3)-phi_x(6)},
{{1,4},phi_x(1)*phi_x(4)},
{{1,5},phi_x(1)*phi_x(5)},
{{1,6},phi_x(1)*phi_x(6)},
{{2,3},phi_x(2)*phi_x(3)},
{{2,4},phi_x(2)*phi_x(4)-phi_x(5)},
{{2,5},phi_x(2)*phi_x(5)},
{{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 \quad \Phi USD is  an automorphism">>$

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

%write "phi:=",phi$

write "USD det \Phi:=",det(phi),"USD"$

write "\\ USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$


J2:=phi**(-1)*MATJ*phi$

for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$

write "\\ The parameters USDxi(1,1), xi(5,5)USD are invariant under automorphisms"$
write "Take the following values :"$

b(2,2):=b(1,1)/xi(1,2)$
write "\\ USD b(2,2):=",b(2,2),"USD"$
write "in order to get USD J2^1_2=1 USD and "$
b(3,3):=b(1,1)**2*xi(3,4)/xi(1,2)$
write "\\ USD b(3,3):=",WS,"USD"$
write "in order to get USD J2^3_4=1 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\\USD J2(3,2):=(b(4,2)*xi(5,5)*xi(3,4)*xi(1,2) + b(4,2)*xi(3,4)*xi(1,2)*xi(1,1)
%+ b(3,2)*xi(1,2)*xi(1,1)**2 + b(3,2)*xi(1,2) - b(3,1)*xi(5,5)*xi(1,2) - b(3,1)*
%xi(1,2)*xi(1,1) + b(1,1)*xi(5,5)*xi(3,2) + b(1,1)*xi(3,2)*xi(1,1))/(b(1,1)**2*xi
%(3,4)*(xi(5,5) + xi(1,1)))USD\\$

b(4,2):=-( b(3,2)*xi(1,2)*xi(1,1)**2 + b(3,2)*xi(1,2) - b(3,1)*xi(5,5)*xi(1,2) - b(3,1)*
xi(1,2)*xi(1,1) + b(1,1)*xi(5,5)*xi(3,2) + b(1,1)*xi(3,2)*xi(1,1))/
(xi(1,2)*xi(3,4)*(xi(5,5) + xi(1,1)))$

write "\\ USD b(4,2):=",WS,"USD"$
write "in order to get USD J2^3_2=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%( - ((b(4,1)*xi(3,4)*xi(1,2) + b(3,2)*xi(1,2)*xi(1,1)**2 + b(3,2)*xi(1,2) - b(1,
%1)*xi(4,2)*xi(3,4))*(xi(5,5) + xi(1,1)) - (b(3,1)*xi(1,2) - b(1,1)*xi(3,2))*
%(2*xi(5,5)*xi(1,1) + xi(1,1)**2 - 1))
%)/
%((xi(5,5) + xi(1,1))*b(1,1)**2*xi(3,4))$

 b(4,1) :=
 ((b(3,1)*xi(1,2) - b(1,1)*xi(3,2))* (2*xi(5,5)*xi(1,1) + xi(1,1)**2 - 1)
/(xi(5,5) + xi(1,1))
 -(+ b(3,2)*xi(1,2)*xi(1,1)**2 + b(3,2)*xi(1,2) - b(1,1)*xi(4,2)*xi(3,4))
)/
 (xi(3,4)*xi(1,2))$
write "\\ USD b(4,1):=",WS,"USD"$
write "in order to get USD J2^4_2=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

b(6,2):=
-( - b(6,1)*xi(5,5)
*xi(3,4)*xi(1,2)*xi(1,1)**2 - b(6,1)*xi(5,5)*xi(3,4)*xi(1,2) - b(6,1)*xi(3,4)*xi
(1,2)*xi(1,1)**3 - b(6,1)*xi(3,4)*xi(1,2)*xi(1,1) - b(5,2)*xi(5,5)**2*xi(3,4)**2
*xi(1,2)**2 - 2*b(5,2)*xi(5,5)*xi(3,4)**2*xi(1,2)**2*xi(1,1) - b(5,2)*xi(3,4)**2
*xi(1,2)**2*xi(1,1)**2 - b(3,2)*xi(6,4)*xi(1,2)*xi(1,1)**4 - 2*b(3,2)*xi(6,4)*xi
(1,2)*xi(1,1)**2 - b(3,2)*xi(6,4)*xi(1,2) + b(3,2)*xi(6,3)*xi(5,5)*xi(3,4)*xi(1,
2)*xi(1,1)**2 + b(3,2)*xi(6,3)*xi(5,5)*xi(3,4)*xi(1,2) + b(3,2)*xi(6,3)*xi(3,4)*
xi(1,2)*xi(1,1)**3 + b(3,2)*xi(6,3)*xi(3,4)*xi(1,2)*xi(1,1) + b(3,1)*xi(6,4)*xi(
5,5)*xi(1,2)*xi(1,1)**2 + b(3,1)*xi(6,4)*xi(5,5)*xi(1,2) + b(3,1)*xi(6,4)*xi(1,2
)*xi(1,1)**3 + b(3,1)*xi(6,4)*xi(1,2)*xi(1,1) - b(1,1)*xi(6,4)*xi(5,5)*xi(3,2)*
xi(1,1)**2 - b(1,1)*xi(6,4)*xi(5,5)*xi(3,2) - b(1,1)*xi(6,4)*xi(3,2)*xi(1,1)**3
- b(1,1)*xi(6,4)*xi(3,2)*xi(1,1) + b(1,1)*xi(6,2)*xi(5,5)*xi(3,4)*xi(1,1)**2 + b
(1,1)*xi(6,2)*xi(5,5)*xi(3,4) + b(1,1)*xi(6,2)*xi(3,4)*xi(1,1)**3 + b(1,1)*xi(6,
2)*xi(3,4)*xi(1,1))/(b(1,1)**3*xi(3,4)**2*(xi(5,5)*xi(1,1)**2 + xi(5,5) + xi(1,1
)**3 + xi(1,1)))
/
((xi(1,2)*( - xi(5,5) + xi(1,1)))/(b(1,1)**3*xi(3,4)))
$
write "\\ USD b(6,2):=",WS,"USD"$
write "in order to get USD J2^6_2=0 USD and "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then one gets for USDJ2USD\\"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence we are led to the case where"$
write "xi(3,2)=xi(4,2)=xi(6,2)= 0 USD: \\"$
write "and USD xi(1,2)=xi(3,4)=1USD:"$
 xi(1,2):=1$
 xi(3,4):=1$
 xi(3,2):=0$
 xi(4,2):=0$
 xi(6,2):=0$

 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
write "\\"$
%write "{fontsize{8}{10} \selectfont"$


 CLEAR b(2,2),b(3,3),b(4,1),b(4,2),b(6,2)$

 write "\par Now we'll use again equivalence by automorphisms"$
 write "starting from the case"$
write "xi(3,2)=xi(4,2)=xi(6,2)= 0 USD: \\"$
write "and USD xi(1,2)=xi(3,4)=1.USD"$
write "We take USD b(2,2)= b(1,1)= b(3,3):=1,"$
write "b(4,2)= b(4,1)= b(3,2)= b(3,1)= b(6,1)= b(5,2)= b(6,2):=0USD"$
write "in order to preserve the foregoing conditions"$
 b(2,2):=1$
 b(1,1):=1$
 b(3,3):=1$
 b(4,2):=0$
 b(4,1):=0$
 b(3,2):=0$
 b(3,1):=0$
 b(6,1):=0$
 b(5,2):=0$
 b(6,2):=0$
write "Take the following values :"$

%\\USD J2(6,1):=( - b(5,1)*xi(5,5) - b(5,1)*xi(1,1) + xi(6,1)*xi(1,1)**2 + xi(6,1
%))/(xi(1,1)**2 + 1)USD\\$
b(5,1):=(
+ xi(6,1)*xi(1,1)**2 + xi(6,1
))/(xi(5,5)+xi(1,1))$
write "\\ USD b(5,1):=",WS,"USD"$
write "in order to get USD J2^6_1=0 USD and "$

%\\USD J2(6,4):= - b(6,3) + xi(6,4)USD\\$
b(6,3) := xi(6,4)$
write "\\ USD b(6,3):=",WS,"USD"$
write "in order to get USD J2^6_4=0 USD and "$

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

b(5,3)
:=( (xi(1,1)**2 + 1)*xi(6,3)*(xi(5,5) + xi(1,1))
- (xi(5,5)**2 + 1)*(xi(1,1)**2 + 1)*xi(6,4))
/ (xi(5,5) + xi(1,1))**2 $
write "\\ USD b(5,3):=",WS,"USD"$
write "in order to get USD J2^6_3=0 USD. "$
write "This second USD\PhiUSD has entries :"$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  \Phi^",i,"_",j,"=",phi(i,j),";USD\\">>$

write "\\ USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$


J2:=phi**(-1)*MATJ*phi$

for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence we are led to the case where"$
write "xi(3,2)=xi(4,2)=xi(6,2)=xi(6,1)=xi(6,3)=xi(6,4)= 0, USD: \\"$
write "USD xi(1,2)=xi(3,4)=1USD"$
write "and USDxi(1,1)=\alpha,xi(5,5) = \beta , \alpha \neq -\beta.USD"$
 xi(1,2):=1$
 xi(3,4):=1$
 xi(3,2):=0$
 xi(4,2):=0$
 xi(6,2):=0$
 xi(6,1):=0$
 xi(6,3):=0$
 xi(6,4):=0$
 xi(1,1):=alpha$
 xi(5,5):=beta$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

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 "\\ 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"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% write "\end{document}"$
% bye;
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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 B(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM JOIN
   FOR j2:=1:DIM JOIN
   IF B(j1,j2) NEQ 0 THEN
   {{{j1,j2},B(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 "Case USD alpha =1 , beta=0USD :"$
alpha:=1$
beta:=0$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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;"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 write "\end{document}"$
 bye;