generic derivation : delta:=
mat((xi(1,1),0,0,0,0,0),(xi(2,1),2*xi(1,1),0,0,0,0),(xi(3,1),xi(3,2),3*xi(1,1),0
,0,0),(xi(4,1),xi(4,2),xi(3,2),4*xi(1,1),0,0),(xi(5,1),xi(5,2),xi(4,2) - xi(3,1)
,xi(3,2) + xi(2,1),5*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),0,0,0,xi(6,6)))$
$


        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  1  0  0  0  0]
adx1 := [                ]
        [0  0  1  0  0  0]
        [                ]
        [0  0  0  1  0  0]
        [                ]
        [0  0  0  0  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
adx2 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  1  0  0  0]
        [                 ]
        [0   0  0  0  0  0]



        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
adx3 := [                  ]
        [-1  0   0  0  0  0]
        [                  ]
        [0   -1  0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
adx4 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(1,1):=0

xi(6,6):=0

by subtracting adjoints one then may suppose:

xi(3,1):=0,xi(3,2):=0,xi(4,1):=0,xi(5,1):=0

delta:=

[   0        0        0        0     0     0   ]
[                                              ]
[xi(2,1)     0        0        0     0     0   ]
[                                              ]
[   0        0        0        0     0     0   ]
[                                              ]
[   0     xi(4,2)     0        0     0     0   ]
[                                              ]
[   0     xi(5,2)  xi(4,2)  xi(2,1)  0  xi(5,6)]
[                                              ]
[xi(6,1)  xi(6,2)     0        0     0     0   ]


We denote this delta by the shortform

shortformdelta:={xi(2,1),

                 ss,

                 xi(4,2),

                 ss,

                 xi(5,2),

                 xi(5,6),

                 ss,

                 xi(6,1),

                 xi(6,2)}

paramindexeslist:={{2,1},{4,2},{5,2},{5,6},{6,1},{6,2}}

delta:=
mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0
,0,0,0))$
$

shortformdelta:={0,ss,0,ss,0,0,ss,1,0}$

on resout l'equation {{0,1},0} qui est  maintenant AA:= - d(0,6)$

Unknown: d(0,6)
Unknown: d(0,6)
bonne inconnue W:=d(0,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,1},1} qui est  maintenant AA:= - d(1,6)$

Unknown: d(1,6)
Unknown: d(1,6)
bonne inconnue W:=d(1,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,1},2} qui est  maintenant AA:= - d(2,6)$

Unknown: d(2,6)
Unknown: d(2,6)
bonne inconnue W:=d(2,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,1},3} qui est  maintenant AA:= - (d(3,6) + d(2,0))$

Unknowns: {d(3,6),d(2,0)}
Unknowns: {d(3,6),d(2,0)}
bonne inconnue W:=d(3,6)$

sa valeur doit etre WW:= - d(2,0)$

on resout l'equation {{0,1},4} qui est  maintenant AA:= - (d(4,6) + d(3,0))$

Unknowns: {d(4,6),d(3,0)}
Unknowns: {d(4,6),d(3,0)}
bonne inconnue W:=d(4,6)$

sa valeur doit etre WW:= - d(3,0)$

on resout l'equation {{0,1},5} qui est  maintenant AA:= - (d(5,6) + d(4,0))$

Unknowns: {d(5,6),d(4,0)}
Unknowns: {d(5,6),d(4,0)}
bonne inconnue W:=d(5,6)$

sa valeur doit etre WW:= - d(4,0)$

on resout l'equation {{0,1},6} qui est  maintenant AA:= - d(6,6) + d(1,1) + d(0,
0)$

Unknowns: {d(6,6),d(1,1),d(0,0)}
Unknowns: {d(6,6),d(1,1),d(0,0)}
bonne inconnue W:=d(6,6)$

sa valeur doit etre WW:=d(1,1) + d(0,0)$

on resout l'equation {{0,2},3} qui est  maintenant AA:=d(1,0)$

Unknown: d(1,0)
Unknown: d(1,0)
bonne inconnue W:=d(1,0)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},5} qui est  maintenant AA:= - d(3,0)$

Unknown: d(3,0)
Unknown: d(3,0)
bonne inconnue W:=d(3,0)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},6} qui est  maintenant AA:=d(1,2)$

Unknown: d(1,2)
Unknown: d(1,2)
bonne inconnue W:=d(1,2)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},5} qui est  maintenant AA:=d(2,0)$

Unknown: d(2,0)
Unknown: d(2,0)
bonne inconnue W:=d(2,0)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},6} qui est  maintenant AA:=d(1,3)$

Unknown: d(1,3)
Unknown: d(1,3)
bonne inconnue W:=d(1,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},6} qui est  maintenant AA:=d(1,4)$

Unknown: d(1,4)
Unknown: d(1,4)
bonne inconnue W:=d(1,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},6} qui est  maintenant AA:=d(1,5)$

Unknown: d(1,5)
Unknown: d(1,5)
bonne inconnue W:=d(1,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},0} qui est  maintenant AA:= - d(0,3)$

Unknown: d(0,3)
Unknown: d(0,3)
bonne inconnue W:=d(0,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},2} qui est  maintenant AA:= - d(2,3)$

Unknown: d(2,3)
Unknown: d(2,3)
bonne inconnue W:=d(2,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},3} qui est  maintenant AA:= - d(3,3) + d(2,2) + d(1,
1)$

Unknowns: {d(3,3),d(2,2),d(1,1)}
Unknowns: {d(3,3),d(2,2),d(1,1)}
bonne inconnue W:=d(3,3)$

sa valeur doit etre WW:=d(2,2) + d(1,1)$

on resout l'equation {{1,2},4} qui est  maintenant AA:= - d(4,3) + d(3,2)$

Unknowns: {d(4,3),d(3,2)}
Unknowns: {d(4,3),d(3,2)}
bonne inconnue W:=d(4,3)$

sa valeur doit etre WW:=d(3,2)$

on resout l'equation {{1,2},5} qui est  maintenant AA:= - d(5,3) + d(4,2) - d(3,
1)$

Unknowns: {d(5,3),d(4,2),d(3,1)}
Unknowns: {d(5,3),d(4,2),d(3,1)}
bonne inconnue W:=d(5,3)$

sa valeur doit etre WW:=d(4,2) - d(3,1)$

on resout l'equation {{1,2},6} qui est  maintenant AA:= - (d(6,3) + d(0,2))$

Unknowns: {d(6,3),d(0,2)}
Unknowns: {d(6,3),d(0,2)}
bonne inconnue W:=d(6,3)$

sa valeur doit etre WW:= - d(0,2)$

on resout l'equation {{1,3},0} qui est  maintenant AA:= - d(0,4)$

Unknown: d(0,4)
Unknown: d(0,4)
bonne inconnue W:=d(0,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},2} qui est  maintenant AA:= - d(2,4)$

Unknown: d(2,4)
Unknown: d(2,4)
bonne inconnue W:=d(2,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},3} qui est  maintenant AA:= - d(3,4)$

Unknown: d(3,4)
Unknown: d(3,4)
bonne inconnue W:=d(3,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},4} qui est  maintenant AA:= - d(4,4) + d(2,2) + 2*d(
1,1)$

Unknowns: {d(4,4),d(2,2),d(1,1)}
Unknowns: {d(4,4),d(2,2),d(1,1)}
bonne inconnue W:=d(4,4)$

sa valeur doit etre WW:=d(2,2) + 2*d(1,1)$

on resout l'equation {{1,3},5} qui est  maintenant AA:= - d(5,4) + d(3,2) + d(2,
1)$

Unknowns: {d(5,4),d(3,2),d(2,1)}
Unknowns: {d(5,4),d(3,2),d(2,1)}
bonne inconnue W:=d(5,4)$

sa valeur doit etre WW:=d(3,2) + d(2,1)$

on resout l'equation {{1,3},6} qui est  maintenant AA:= - d(6,4)$

Unknown: d(6,4)
Unknown: d(6,4)
bonne inconnue W:=d(6,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},0} qui est  maintenant AA:= - d(0,5)$

Unknown: d(0,5)
Unknown: d(0,5)
bonne inconnue W:=d(0,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},2} qui est  maintenant AA:= - d(2,5)$

Unknown: d(2,5)
Unknown: d(2,5)
bonne inconnue W:=d(2,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},3} qui est  maintenant AA:= - d(3,5)$

Unknown: d(3,5)
Unknown: d(3,5)
bonne inconnue W:=d(3,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},4} qui est  maintenant AA:= - d(4,5)$

Unknown: d(4,5)
Unknown: d(4,5)
bonne inconnue W:=d(4,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},5} qui est  maintenant AA:= - d(5,5) + d(2,2) + 3*d(
1,1)$

Unknowns: {d(5,5),d(2,2),d(1,1)}
Unknowns: {d(5,5),d(2,2),d(1,1)}
bonne inconnue W:=d(5,5)$

sa valeur doit etre WW:=d(2,2) + 3*d(1,1)$

on resout l'equation {{1,4},6} qui est  maintenant AA:= - d(6,5)$

Unknown: d(6,5)
Unknown: d(6,5)
bonne inconnue W:=d(6,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{2,3},5} qui est  maintenant AA:=d(2,2) - 2*d(1,1)$

Unknowns: {d(2,2),d(1,1)}
Unknowns: {d(2,2),d(1,1)}
bonne inconnue W:=d(2,2)$

sa valeur doit etre WW:=2*d(1,1)$

Derivation equations to cancel (Reduce output) : \\{{{{0,1},0},0},
{{{0,1},1},0},
{{{0,1},2},0},
{{{0,1},3},0},
{{{0,1},4},0},
{{{0,1},5},0},
{{{0,1},6},0},
{{{0,2},3},0},
{{{0,2},5},0},
{{{0,2},6},0},
{{{0,3},4},0},
{{{0,3},5},0},
{{{0,3},6},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},6},0},
{{{0,6},6},0},
{{{1,2},0},0},
{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},0},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},0},
{{{1,3},0},0},
{{{1,3},1},0},
{{{1,3},2},0},
{{{1,3},3},0},
{{{1,3},4},0},
{{{1,3},5},0},
{{{1,3},6},0},
{{{1,4},0},0},
{{{1,4},1},0},
{{{1,4},2},0},
{{{1,4},3},0},
{{{1,4},4},0},
{{{1,4},5},0},
{{{1,4},6},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,5},5},0},
{{{1,5},6},0},
{{{1,6},3},0},
{{{1,6},4},0},
{{{1,6},5},0},
{{{1,6},6},0},
{{{2,3},0},0},
{{{2,3},1},0},
{{{2,3},2},0},
{{{2,3},3},0},
{{{2,3},4},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},3},0},
{{{2,4},5},0},
{{{2,5},3},0},
{{{2,5},5},0},
{{{2,6},3},0},
{{{2,6},5},0},
{{{3,4},4},0},
{{{3,4},5},0},
{{{3,5},4},0},
{{{3,5},5},0},
{{{3,6},4},0},
{{{3,6},5},0},
{{{4,5},5},0},
{{{4,6},5},0}}$

Il n'y a pas de phase 2$

derivation generique de gtildedelta:$

MATD:=
mat((d(0,0),d(0,1),d(0,2),0,0,0,0),(0,d(1,1),0,0,0,0,0),(0,d(2,1),2*d(1,1),0,0,0
,0),(0,d(3,1),d(3,2),3*d(1,1),0,0,0),(d(4,0),d(4,1),d(4,2),d(3,2),4*d(1,1),0,0),
(d(5,0),d(5,1),d(5,2),d(4,2) - d(3,1),d(3,2) + d(2,1),5*d(1,1), - d(4,0)),(d(6,0
),d(6,1),d(6,2), - d(0,2),0,0,d(1,1) + d(0,0)))$
$

pour delta:=

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


pour shortformdelta:={0,ss,0,ss,0,0,ss,1,0}

Unknowns: {d(6,2),
d(6,1),
d(6,0),
d(5,2),
d(5,1),
d(5,0),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,1),
d(2,1),
d(1,1),
d(0,2),
d(0,1),
d(0,0)}
Unknowns: {d(6,2),
d(6,1),
d(6,0),
d(5,2),
d(5,1),
d(5,0),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,1),
d(2,1),
d(1,1),
d(0,2),
d(0,1),
d(0,0)}
listeparametresMATD{d(6,2),
d(6,1),
d(6,0),
d(5,2),
d(5,1),
d(5,0),
d(4,2),
d(4,1),
d(4,0),
d(3,2),
d(3,1),
d(2,1),
d(1,1),
d(0,2),
d(0,1),
d(0,0)}$

dim Der(gtildedelta):=16$

t1:=D(0,0):=

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


Unknowns: {d(6,0),

           d(5,2),

           d(5,1),

           d(4,2),

           d(4,1),

           d(3,2),

           d(3,1),

           d(2,1),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,0),

           d(5,2),

           d(5,1),

           d(4,2),

           d(4,1),

           d(3,2),

           d(3,1),

           d(2,1),

           d(1,1),

           d(0,0)}

commutant de t1 dans der(gtildedelta):


mat((d(0,0),0,0,0,0,0,0),

    (0,d(1,1),0,0,0,0,0),

    (0,d(2,1),2*d(1,1),0,0,0,0),

    (0,d(3,1),d(3,2),3*d(1,1),0,0,0),

    (0,d(4,1),d(4,2),d(3,2),4*d(1,1),0,0),

    (0,d(5,1),d(5,2),d(4,2) - d(3,1),d(3,2) + d(2,1),5*d(1,1),0),

    (d(6,0),0,0,0,0,0,d(1,1) + d(0,0)))


Unknowns: {d(6,0),

           d(5,2),

           d(5,1),

           d(4,2),

           d(4,1),

           d(3,2),

           d(3,1),

           d(2,1),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,0),

           d(5,2),

           d(5,1),

           d(4,2),

           d(4,1),

           d(3,2),

           d(3,1),

           d(2,1),

           d(1,1),

           d(0,0)}

t2:=D(1,1):=

[0  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  2  0  0  0  0]
[                   ]
[0  0  0  3  0  0  0]
[                   ]
[0  0  0  0  4  0  0]
[                   ]
[0  0  0  0  0  5  0]
[                   ]
[0  0  0  0  0  0  1]


Unknowns: {d(1,1),d(0,0)}
Unknowns: {d(1,1),d(0,0)}
commutant simultane de t1,t2 dans der(gtildedelta):$


mat((d(0,0),0,0,0,0,0,0),(0,d(1,1),0,0,0,0,0),(0,0,2*d(1,1),0,0,0,0),(0,0,0,3*d(
1,1),0,0,0),(0,0,0,0,4*d(1,1),0,0),(0,0,0,0,0,5*d(1,1),0),(0,0,0,0,0,0,d(1,1) + 
d(0,0)))$
$

rank 2 with maximal torus t1,t2

2


matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base  diagonali\
sant le tore maximal: on peut prendre

P:=

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


P**(-1)*t1*P:=

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

P**(-1)*t2*P:=

[0  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  2  0  0  0  0]
[                   ]
[0  0  0  3  0  0  0]
[                   ]
[0  0  0  0  4  0  0]
[                   ]
[0  0  0  0  0  5  0]
[                   ]
[0  0  0  0  0  0  1]


matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):

P**(-1)*MATD*P:=
mat((d(0,0),d(0,1),d(0,2),0,0,0,0),(0,d(1,1),0,0,0,0,0),(0,d(2,1),2*d(1,1),0,0,0
,0),(0,d(3,1),d(3,2),3*d(1,1),0,0,0),(d(4,0),d(4,1),d(4,2),d(3,2),4*d(1,1),0,0),
(d(5,0),d(5,1),d(5,2),d(4,2) - d(3,1),d(3,2) + d(2,1),5*d(1,1), - d(4,0)),(d(6,0
),d(6,1),d(6,2), - d(0,2),0,0,d(1,1) + d(0,0)))$
$

PP:=

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


avec PP:=P*Q:=

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


MATDDIAGONALISE:=

mat((d(0,0),d(0,1),d(0,2),0,0,0,0),

    (0,d(1,1),0,0,0,0,0),

    (0,d(2,1),2*d(1,1),0,0,0,0),

    (0,d(3,1),d(3,2),3*d(1,1),0,0,0),

    (d(4,0),d(4,1),d(4,2),d(3,2),4*d(1,1),0,0),

    (d(5,0),d(5,1),d(5,2),d(4,2) - d(3,1),d(3,2) + d(2,1),5*d(1,1), - d(4,0)),

    (d(6,0),d(6,1),d(6,2), - d(0,2),0,0,d(1,1) + d(0,0)))


on voit apparaitre les poids sur la diagonale

*** r declared operator 

r(1) := d(0,0)

r(2) := d(1,1)

r(3) := 2*d(1,1)

r(4) := 3*d(1,1)

r(5) := 4*d(1,1)

r(6) := 5*d(1,1)

r(7) := d(1,1) + d(0,0)

r(1) := gamma2

r(2) := gamma1

r(3) := 2*gamma1

r(4) := 3*gamma1

r(5) := 4*gamma1

r(6) := 5*gamma1

r(7) := gamma1 + gamma2

Le systeme de poids est le systeme 2.20

calcul de relations de commutation de la base diaY(j) diagonalisant le tore

listcommutateursdesx := {{{0,1},x(6)},

   {{0,2},0},

   {{0,3},0},

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

   {{1,2},x(3)},

   {{1,3},x(4)},

   {{1,4},x(5)},

   {{1,5},0},

   {{1,6},0},

   {{2,3},x(5)},

   {{2,4},0},

   {{2,5},0},

   {{2,6},0},

   {{3,4},0},

   {{3,5},0},

   {{3,6},0},

   {{4,5},0},

   {{4,6},0},

   {{5,6},0}}

diaY(1):=x(0)

diaY(2):=x(1)

diaY(3):=x(2)

diaY(4):=x(3)

diaY(5):=x(4)

diaY(6):=x(5)

diaY(7):=x(6)

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2},diay(7)},
{{1,3},0},
{{1,4},0},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diay(4)},
{{2,4},diay(5)},
{{2,5},diay(6)},
{{2,6},0},
{{2,7},0},
{{3,4},diay(6)},
{{3,5},0},
{{3,6},0},
{{3,7},0},
{{4,5},0},
{{4,6},0},
{{4,7},0},
{{5,6},0},
{{5,7},0},
{{6,7},0}}$

Now we make explicit the isomorphism with an algebra of the book:$

Namely g_{7,2.20}$

i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:$

on pose :$

avec comme matrice de changement de base :$

This isomorphism has been computed by calculisom6_56xCVIII.red$


mat((0,1,0,0,0,0,0),(1,0,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,0,1,0,0),(0,0,0,0,0,1
,0),(0,0,0,0,0,0,1),(0,0,0,-1,0,0,0))$
$

det(isom):= -1$

ZZ(1):=diay(2)$

ZZ(2):=diay(1)$

ZZ(3):=diay(3)$

ZZ(4):= - diay(7)$

ZZ(5):=diay(4)$

ZZ(6):=diay(5)$

ZZ(7):=diay(6)$

listcommutateursdesZZ:=$

{{1,2},zz(4)}$

{{1,3},zz(5)}$

{{1,4},0}$

{{1,5},zz(6)}$

{{1,6},zz(7)}$

{{1,7},0}$

{{2,3},0}$

{{2,4},0}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},0}$

{{3,5},zz(7)}$

{{3,6},0}$

{{3,7},0}$

{{4,5},0}$

{{4,6},0}$

{{4,7},0}$

{{5,6},0}$

{{5,7},0}$

{{6,7},0}$

We get the commutation relations of$

g_{7,2.20}$

shortformdelta:={0,ss,0,ss,0,0,ss,1,0}$

delta:=
mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,0,0),(1,0,0
,0,0,0))$
$