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


        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
adx1 := [                ]
        [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  0]
adx2 := [                 ]
        [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  0]
adx3 := [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  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]
adx4 := [                 ]
        [0  0  0   0  0  0]
        [                 ]
        [0  0  0   0  0  0]
        [                 ]
        [0  0  -1  0  0  0]


The generic nilpotent derivation : 

the matrix A:=((xi(1,1),xi(1,2)),xi(2,1),xi(2,2))) has to be nilpotent

the matrix B:=((xi(3,3),xi(3,4)),xi(4,3),xi(4,4))) has to be nilpotent

By Jordan reduction with a first kind automorphism, we get 4 cases

Case 1 : A:=0, B:=0

Case 2 : A:=((0,1),(0,0)), B:=0

Case 22 : A:=0, B:=((0,1),(0,0))

Case 3 : A:= B:=((0,1),(0,0))

The cases 2 and 22 are intertwined by some second kind automorphism

Hence we get only three case 1,2,3.

We consider here the case 1.

Then A:=B:=0

xi(1,1):=0

xi(1,2):=0

xi(2,1):=0

xi(2,2):=0

xi(3,3):=0

xi(3,4):=0

xi(4,3):=0

xi(4,4):=0

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]
[                                        ]
[xi(5,1)  xi(5,2)  xi(5,3)  xi(5,4)  0  0]
[                                        ]
[xi(6,1)  xi(6,2)  xi(6,3)  xi(6,4)  0  0]


by subtracting adjoints one then may suppose:

xi(5,1):=0,xi(5,2):=0,xi(6,3):=0,xi(6,4):=0

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     xi(5,3)  xi(5,4)  0  0]
[                                        ]
[xi(6,1)  xi(6,2)     0        0     0  0]


We denote this delta by the shortform

shortformdelta:={xi(5,3),

                 xi(5,4),

                 ss,

                 xi(6,1),

                 xi(6,2)}

paramindexeslist:={{5,3},{5,4},{6,1},{6,2}}

a:=0$

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


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

on resout l'equation {{0,1},5} qui est  maintenant AA:=d(3,1) - d(2,0)$

Unknowns: {d(3,1),d(2,0)}
Unknowns: {d(3,1),d(2,0)}
bonne inconnue W:=d(3,1)$

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

on resout l'equation {{0,2},5} qui est  maintenant AA:=d(3,2) + d(1,0)$

Unknowns: {d(3,2),d(1,0)}
Unknowns: {d(3,2),d(1,0)}
bonne inconnue W:=d(3,2)$

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

on resout l'equation {{0,3},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 {{0,3},1} 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 {{0,3},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 {{0,3},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 {{0,3},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 {{0,3},5} qui est  maintenant AA:= - d(5,5) + d(3,3) + d(0,
0)$

Unknowns: {d(5,5),d(3,3),d(0,0)}
Unknowns: {d(5,5),d(3,3),d(0,0)}
bonne inconnue W:=d(5,5)$

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

on resout l'equation {{0,3},6} qui est  maintenant AA:= - (d(6,5) + d(4,0))$

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

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

on resout l'equation {{0,4},5} 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 {{0,4},6} 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 {{0,6},5} qui est  maintenant AA:=d(3,6)$

Unknown: d(3,6)
Unknown: d(3,6)
bonne inconnue W:=d(3,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},5} qui est  maintenant AA:= - d(3,3) + d(2,2) + d(1,
1) - d(0,0)$

Unknowns: {d(3,3),d(2,2),d(1,1),d(0,0)}
Unknowns: {d(3,3),d(2,2),d(1,1),d(0,0)}
bonne inconnue W:=d(3,3)$

sa valeur doit etre WW:=d(2,2) + d(1,1) - d(0,0)$

on resout l'equation {{1,2},6} qui est  maintenant AA:=d(4,0)$

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

sa valeur doit etre WW:=0$

on resout l'equation {{1,3},5} qui est  maintenant AA:=d(2,3) + d(0,1)$

Unknowns: {d(2,3),d(0,1)}
Unknowns: {d(2,3),d(0,1)}
bonne inconnue W:=d(2,3)$

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

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

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

sa valeur doit etre WW:=0$

on resout l'equation {{1,4},5} 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 {{1,4},6} 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 {{1,6},5} 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 {{2,3},5} qui est  maintenant AA:= - d(1,3) + d(0,2)$

Unknowns: {d(1,3),d(0,2)}
Unknowns: {d(1,3),d(0,2)}
bonne inconnue W:=d(1,3)$

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

on resout l'equation {{2,3},6} qui est  maintenant AA:= - d(4,2)$

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

sa valeur doit etre WW:=0$

on resout l'equation {{2,4},5} 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 {{2,4},6} 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 {{2,6},5} 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 {{3,4},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 {{3,4},4} qui est  maintenant AA:= - d(4,6)$

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

sa valeur doit etre WW:=0$

on resout l'equation {{3,4},5} qui est  maintenant AA:= - (d(5,6) + d(0,4))$

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

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

on resout l'equation {{3,4},6} qui est  maintenant AA:= - d(6,6) + d(4,4) + d(2,
2) + d(1,1) - d(0,0)$

Unknowns: {d(6,6),d(4,4),d(2,2),d(1,1),d(0,0)}
Unknowns: {d(6,6),d(4,4),d(2,2),d(1,1),d(0,0)}
bonne inconnue W:=d(6,6)$

sa valeur doit etre WW:=d(4,4) + d(2,2) + d(1,1) - d(0,0)$

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

Il n'y a pas de phase 2$

derivation generique de gtildedelta:$

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


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

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

dim Der(gtildedelta):=21$

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  -1  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(6,3),

           d(5,4),

           d(5,2),

           d(5,1),

           d(4,4),

           d(2,2),

           d(2,1),

           d(1,2),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,3),

           d(5,4),

           d(5,2),

           d(5,1),

           d(4,4),

           d(2,2),

           d(2,1),

           d(1,2),

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

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

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

    (0,0,0,0,d(4,4),0,0),

    (0,d(5,1),d(5,2),0,d(5,4),d(2,2) + d(1,1),0),

    (0,0,0,d(6,3),0,0,d(4,4) + d(2,2) + d(1,1) - d(0,0)))


Unknowns: {d(6,3),

           d(5,4),

           d(5,2),

           d(5,1),

           d(4,4),

           d(2,2),

           d(2,1),

           d(1,2),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,3),

           d(5,4),

           d(5,2),

           d(5,1),

           d(4,4),

           d(2,2),

           d(2,1),

           d(1,2),

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


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

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

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

t3:=D(2,2):=

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


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

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

commutant simultane de t1,t2,t3 dans der(gtildedelta):


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

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

    (0,0,d(2,2),0,0,0,0),

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

    (0,0,0,0,d(4,4),0,0),

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

    (0,0,0,d(6,3),0,0,d(4,4) + d(2,2) + d(1,1) - d(0,0)))


MATD:=

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

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

    (0,0,d(2,2),0,0,0,0),

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

    (0,0,0,0,d(4,4),0,0),

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

    (0,0,0,d(6,3),0,0,d(4,4) + d(2,2) + d(1,1) - d(0,0)))


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

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

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

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

t4:=D(4,4):=

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


Unknowns: {d(4,4),d(2,2),d(1,1),d(0,0)}

Unknowns: {d(4,4),d(2,2),d(1,1),d(0,0)}

commutant simultane de t1,t2,t3,t4 dans der(gtildedelta):


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

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

    (0,0,d(2,2),0,0,0,0),

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

    (0,0,0,0,d(4,4),0,0),

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

    (0,0,0,0,0,0,d(4,4) + d(2,2) + d(1,1) - d(0,0)))


rank 4 with maximal torus t1,t2,t3,t4

4


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

P**(-1)*t3*P:=

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

P**(-1)*t4*P:=

[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  0  0]
[                   ]
[0  0  0  0  0  0  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),d(0,3),d(0,4),0,0),(0,d(1,1),d(1,2),d(0,2),0,0,0),(0,d
(2,1),d(2,2), - d(0,1),0,0,0),(0,0,0,d(2,2) + d(1,1) - d(0,0),0,0,0),(0,0,0,d(4,
3),d(4,4),0,0),(d(5,0),d(5,1),d(5,2),d(5,3),d(5,4),d(2,2) + d(1,1), - d(0,4)),(d
(6,0),d(6,1),d(6,2),d(6,3),d(6,4),0,d(4,4) + d(2,2) + 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),d(0,3),d(0,4),0,0),

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

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

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

    (0,0,0,d(4,3),d(4,4),0,0),

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

    (d(6,0),d(6,1),d(6,2),d(6,3),d(6,4),0,d(4,4) + d(2,2) + 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) := d(2,2)

r(4) := d(2,2) + d(1,1) - d(0,0)

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

r(6) := d(2,2) + d(1,1)

r(7) := d(4,4) + d(2,2) + d(1,1) - d(0,0)

r(1) := gamma3

r(2) := gamma1

r(3) := gamma2

r(4) := gamma1 - gamma3 + gamma2

r(5) := gamma4

r(6) := gamma1 + gamma2

r(7) := gamma1 - gamma3 + gamma2 + gamma4

Le systeme de poids est le systeme 4.3

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

listcommutateursdesx := {{{0,1},0},

   {{0,2},0},

   {{0,3},x(5)},

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

   {{1,2},x(5)},

   {{1,3},0},

   {{1,4},0},

   {{1,5},0},

   {{1,6},0},

   {{2,3},0},

   {{2,4},0},

   {{2,5},0},

   {{2,6},0},

   {{3,4},x(6)},

   {{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},0},
{{1,3},0},
{{1,4},diay(6)},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3},diay(6)},
{{2,4},0},
{{2,5},0},
{{2,6},0},
{{2,7},0},
{{3,4},0},
{{3,5},0},
{{3,6},0},
{{3,7},0},
{{4,5},diay(7)},
{{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_{7,4.3}$

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 :$


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

det(isom):= 1$

ZZ(1):=diay(2)$

ZZ(2):=diay(3)$

ZZ(3):=diay(1)$

ZZ(4):=diay(5)$

ZZ(5):=diay(4)$

ZZ(6):=diay(6)$

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

listcommutateursdesZZ:=$

{{1,2},zz(6)}$

{{1,3},0}$

{{1,4},0}$

{{1,5},0}$

{{1,6},0}$

{{1,7},0}$

{{2,3},0}$

{{2,4},0}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},0}$

{{3,5},zz(6)}$

{{3,6},0}$

{{3,7},0}$

{{4,5},zz(7)}$

{{4,6},0}$

{{4,7},0}$

{{5,6},0}$

{{5,7},0}$

{{6,7},0}$

We get the commutation relations of$

g_{7,4.3}$

Et cela pour a:=0$

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

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),(0,0,1,0,0,0),(0,0,0
,0,0,0))$
$