%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This is "2009structcomplsl2xsl2bis.red"    :
%renamed as:
%CSsl22.red

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  PRECISER SORTIE RESULTATS   %%%%%%%%%%%%%%%%%%%%%
%ON FACTOR$
   off echo$
    off nat$
%OUT "2009rstructcomplsl2xsl2bis.r"    $
OUT "rCSsl22.r"    $
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "sl2bisxsl2bis.red"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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{CSsl22.red}.\\"$
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"$
%FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE OTEZERO(U)$  %enleve les 0 dans une liste
BEGIN$
LIST U$
RETURN FOR EACH A IN U JOIN IF A NEQ 0 THEN {A} ELSE {}$
END$

PROCEDURE CALLLET(U,V)$
LET U=V$

PROCEDURE UNKNOWNS(U)$
BEGIN$ LIST U$
RETURN  IF U={} THEN U ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE SSOLVE(n,p)$
BEGIN INTEGER k,j$
 k:=1$ j:=1$
S1: IF z(k,j) NEQ 0 THEN <<GO TO S2>> ELSE <<GO TO S3>>$
S2: CALLLET(z(k,j),0) $
S3: IF j<p THEN j:=j+1 ELSE IF k<n  THEN <<k:=k+1$ j:=1>> ELSE GO TO
 S4$
GO TO S1$
S4: END$

PROCEDURE distinct(U)$
BEGIN;INTEGER j;LIST UU$ j:=1;UU:={}$
S: IF U NEQ {} THEN    ZZZ:=PART(u,j)$
IF ZZZ MEMBER UU THEN <<GO TO P>> ELSE UU:= ZZZ. UU $
P: IF j<LENGTH(U) THEN <<j:=j+1,GO TO S>> ELSE <<RETURN REVERSE(UU)>>$
CLEAR ZZZ$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%collecting the torsion equations%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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))$


 for  all a,b such that freeof(b,X) let f(a/b)=f(a)/b$
 for  all a,b such that freeof(b,X) let f(b*a)=b*f(a)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%FOR j:=1:DIM COLLECT X(j)$
%FOR EACH W IN WS DO <<FACTOR W>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE LOCALRECAP$
BEGIN$
write "localrecap"$
WRITE "USD \par Now the nonzero torsion equations left are :"$

write COLLECT_EQ$

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)$

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

 MATJCARRE:=(MATJ)**2$

for i:=1:6 do for j:=1:6 do << write "\\USD J^2(", i ,",", j, "):=" , MATJCARRE(i,j),"USD\\">>$
% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  J^2(",i,",",j,"):=",MATJCARRE(i,j),";USD\\">>$

write "Trace(J):=", trace(matJ)$
on nat$ write "J:=",matJ; off nat$
on nat$ write "J**2:=",matJcarre; off nat$


on factor$
write "det(J):=", det(matJ)$
off factor$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%Computing the complex structures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "************************************************************************"$
write "Here we suppose that:"$
xi(1,1):=0$
xi(1,2):=0$
xi(1,3):=-1$
xi(2,1):=0$

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(5,6):=0$
xi(6,4):=1$
xi(6,5):=0$
xi(6,6):=0$

xi(5,5):=-xi(2,2)$
for i:=1:3 do for j:=1:3 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
for i:=4:6 do for j:=4:6 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
write "************************************************************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "From the equation"$
%\\USD J^2(1,4):= - (xi(3,4) - xi(1,6))USD\\
write "%\\USD J^2(1,4):= - (xi(3,4) - xi(1,6))USD\\"$
xi(3,4) := xi(1,6)$
write "\\ USD xi(3,4):=", ws,"USD"$


write "From the equation"$
%\\USD J^2(1,5):= - (xi(3,5) + xi(2,2)*xi(1,5))USD\\
write "%\\USD J^2(1,5):= - (xi(3,5) + xi(2,2)*xi(1,5))USD\\"$
xi(3,5) :=- xi(2,2)*xi(1,5)$
write "\\ USD xi(3,5):=", ws,"USD"$


write "From the equation"$
%\\USD J^2(1,6):= - (xi(3,6) + xi(1,4))USD\\
write "%\\USD J^2(1,6):= - (xi(3,6) + xi(1,4))USD\\"$
xi(3,6) :=- xi(1,4)$
write "\\ USD xi(3,6):=", ws,"USD"$


write "From the equation"$
%\\USD J^2(2,6):=xi(2,6)*xi(2,2) - xi(2,4)USD\\
write "%\\USD J^2(2,6):=xi(2,6)*xi(2,2) - xi(2,4)USD\\"$
xi(2,4):=xi(2,6)*xi(2,2) $
write "\\ USD xi(2,4):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$
%{{{4,6},2},xi(1,6)**2 + xi(1,4)**2},
write "%{{{4,6},2},xi(1,6)**2 + xi(1,4)**2},"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", ws,"USD"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", ws,"USD"$

write "From the equation"$
%\\USD J^2(2,4):=(xi(2,2)**2 + 1)*xi(2,6)USD\\
write"%\\USD J^2(2,4):=(xi(2,2)**2 + 1)*xi(2,6)USD\\"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", ws,"USD"$


write "From the equation"$
%\\USD J^2(3,5):=(xi(2,2)**2 + 1)*xi(1,5)USD\\
write "%\\USD J^2(3,5):=(xi(2,2)**2 + 1)*xi(1,5)USD\\"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", ws,"USD"$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$

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


write "From the equation"$
%\\USD J^2(4,2):= - (xi(6,2) - xi(4,2)*xi(2,2))USD\\
write "%\\USD J^2(4,2):= - (xi(6,2) - xi(4,2)*xi(2,2))USD\\"$
xi(6,2) := xi(4,2)*xi(2,2)$
write "\\ USD xi(6,2):=", ws,"USD"$



write "From the equation"$
%\\USD J^2(4,3):= - (xi(6,3) + xi(4,1))USD\\
write "%\\USD J^2(4,3):= - (xi(6,3) + xi(4,1))USD\\"$
xi(6,3) :=- xi(4,1)$
write "\\ USD xi(6,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$
%{{{1,3},5},xi(4,3)**2 + xi(4,1)**2},
write "%{{{1,3},5},xi(4,3)**2 + xi(4,1)**2},"$
xi(4,3):=0$
write "\\ USD xi(4,3):=", ws,"USD"$
xi(4,1):=0$
write "\\ USD xi(4,1):=", ws,"USD"$
write "From the equation"$
%{{{2,5},4}, - (xi(2,2)**2 + 1)*xi(4,2)},
write "%{{{2,5},4}, - (xi(2,2)**2 + 1)*xi(4,2)},"$
xi(4,2):=0$
write "\\ USD xi(4,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$
%{{{1,2},5},xi(5,3)*xi(2,2) - xi(5,1)},
write "%{{{1,2},5},xi(5,3)*xi(2,2) - xi(5,1)},"$
xi(5,1):=xi(5,3)*xi(2,2)$
write "\\ USD xi(5,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$
%{{{2,3},5},(xi(2,2)**2 + 1)*xi(5,3)},
write "%{{{2,3},5},(xi(2,2)**2 + 1)*xi(5,3)},"$
xi(5,3):=0$
write "\\ USD xi(5,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Now, From the equation"$
%\\USD J^2(2,2):=xi(5,2)*xi(2,5) + xi(2,2)**2USD\\
write "%\\USD J^2(2,2):=xi(5,2)*xi(2,5) + xi(2,2)**2USD\\"$
xi(5,2):=(-1- xi(2,2)**2)/xi(2,5)$
write "\\ USD xi(5,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
localrecap()$
%bye$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ Then USD J USD has entries :"$
off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 write "Hence we are finally reduced to "$


 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
 write "USDUSD J(\xi^2_2,\xi^2_5) := \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 "with condition USD xi(2,5) \neq 0 USD."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
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-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 "\end{document}"$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%