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

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


The generic nilpotent derivation : the eigenvalues are 0

xi(3,3):= - xi(1,1)

xi(6,6):=0

And the matrix A:=

[xi(1,1)  xi(1,2)   xi(1,3)     xi(1,4)  ]
[                                        ]
[xi(2,1)  xi(2,2)   xi(1,4)     xi(2,4)  ]
[                                        ]
[xi(3,1)  xi(3,2)   - xi(1,1)   - xi(2,1)]
[                                        ]
[xi(3,2)  xi(4,2)   - xi(1,2)   - xi(2,2)]


is nilpotent.

     [xi(1,1)  xi(1,2)   xi(1,3)     xi(1,4)  ]
     [                                        ]
     [xi(2,1)  xi(2,2)   xi(1,4)     xi(2,4)  ]
m := [                                        ]
     [xi(3,1)  xi(3,2)   - xi(1,1)   - xi(2,1)]
     [                                        ]
     [xi(3,2)  xi(4,2)   - xi(1,2)   - xi(2,2)]


we may suppose that A is an element of sp(4,C)+, that is:

xi(1,1):=0

xi(2,1):=0

xi(2,2):=0

xi(3,1):=0

xi(3,2):=0

xi(4,2):=0

M**1:=

[0  xi(1,2)   xi(1,3)    xi(1,4)]
[                               ]
[0     0      xi(1,4)    xi(2,4)]
[                               ]
[0     0         0          0   ]
[                               ]
[0     0      - xi(1,2)     0   ]


M**2:=

[0  0          0           xi(2,4)*xi(1,2)]
[                                         ]
[0  0   - xi(2,4)*xi(1,2)         0       ]
[                                         ]
[0  0          0                  0       ]
[                                         ]
[0  0          0                  0       ]


M**3:=

[                        2   ]
[0  0   - xi(2,4)*xi(1,2)   0]
[                            ]
[0  0           0           0]
[                            ]
[0  0           0           0]
[                            ]
[0  0           0           0]


M**4:=

[0  0  0  0]
[          ]
[0  0  0  0]
[          ]
[0  0  0  0]
[          ]
[0  0  0  0]


Trace(M**1)/2:=0$

Trace(M**2)/2:=0$

Trace(M**3)/2:=0$

Trace(M**4)/2:=0$

************************************************************************$

*******Suppose xi(1,2) neq 0.$

************************************************************************$

By subtracting adjoints one then may suppose:$

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

delta:=

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


We denote this delta by the shortform$

shortformdelta:={xi(1,2),
xi(1,3),
xi(1,4),
ss,
xi(2,4),
ss,
xi(5,6),
ss,
xi(6,1),
xi(6,2),
xi(6,3),
xi(6,4)}$

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

a:=0$

b:=b$

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]
[                ]
[0  1  0  0  0  0]


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

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

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

sa valeur doit etre WW:=0$

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

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

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

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

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

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

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

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

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

on resout l'equation {{2,4},6} 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 {{3,4},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$

Derivation equations to cancel (Reduce output) : \\{{{{0,1},5},0},
{{{0,1},6},0},
{{{0,2},0},0},
{{{0,2},1},0},
{{{0,2},2},0},
{{{0,2},3},0},
{{{0,2},4},0},
{{{0,2},5},0},
{{{0,2},6},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},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},5},0},
{{{1,5},5},0},
{{{1,6},5},0},
{{{2,3},5},0},
{{{2,3},6},0},
{{{2,4},0},0},
{{{2,4},1},0},
{{{2,4},2},0},
{{{2,4},3},0},
{{{2,4},4},0},
{{{2,4},5},0},
{{{2,4},6},0},
{{{2,5},5},0},
{{{2,5},6},0},
{{{2,6},5},0},
{{{2,6},6},0},
{{{3,4},5},0},
{{{3,5},5},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),0,d(0,2),0,0,0,0),(0,d(1,1),d(1,2),d(1,3),0,0,0),(0,0,d(2,2),0,0,0,0
),(0,d(3,1),d(3,2),d(3,3),0,0,0),(d(4,0),d(3,2),d(4,2), - d(1,2),d(3,3) - d(2,2)
 + d(1,1),0,0),(d(5,0),d(5,1),d(5,2),d(5,3),d(5,4),d(3,3) + d(1,1), - d(4,0)),(d
(6,0),d(6,1),d(6,2),d(6,3),d(6,4),0,d(2,2) + 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]
[                ]
[0  1  0  0  0  0]


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

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

           d(5,3),

           d(5,2),

           d(5,1),

           d(4,2),

           d(3,3),

           d(3,2),

           d(3,1),

           d(2,2),

           d(1,3),

           d(1,2),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,0),

           d(5,4),

           d(5,3),

           d(5,2),

           d(5,1),

           d(4,2),

           d(3,3),

           d(3,2),

           d(3,1),

           d(2,2),

           d(1,3),

           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),d(1,3),0,0,0),

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

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

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

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

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


Unknowns: {d(6,0),

           d(5,4),

           d(5,3),

           d(5,2),

           d(5,1),

           d(4,2),

           d(3,3),

           d(3,2),

           d(3,1),

           d(2,2),

           d(1,3),

           d(1,2),

           d(1,1),

           d(0,0)}

Unknowns: {d(6,0),

           d(5,4),

           d(5,3),

           d(5,2),

           d(5,1),

           d(4,2),

           d(3,3),

           d(3,2),

           d(3,1),

           d(2,2),

           d(1,3),

           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  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]


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

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

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

Unknowns: {d(5,1),d(3,3),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(3,3),0,0,0),

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

    (0,d(5,1),0,0,0,d(3,3) + d(1,1),0),

    (0,0,0,0,0,0,d(2,2) + 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(3,3),0,0,0),

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

    (0,d(5,1),0,0,0,d(3,3) + d(1,1),0),

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


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

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

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

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

t4:=D(3,3):=

[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  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  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)}

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(3,3),0,0,0),

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

    (0,0,0,0,0,d(3,3) + d(1,1),0),

    (0,0,0,0,0,0,d(2,2) + 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  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  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]

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  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]

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


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

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

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

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

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

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

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

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


on voit apparaitre les poids sur la diagonale

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

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

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

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

r(5) := d(3,3) - d(2,2) + d(1,1)

r(6) := d(3,3) + d(1,1)

r(7) := d(2,2) + d(0,0)

r(1) := gamma4

r(2) := gamma2

r(3) := gamma1 - gamma3 + gamma2

r(4) := gamma1

r(5) := gamma3

r(6) := gamma1 + gamma2

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

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},x(6)},

   {{0,3},0},

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

   {{1,2},0},

   {{1,3},x(5)},

   {{1,4},0},

   {{1,5},0},

   {{1,6},0},

   {{2,3},0},

   {{2,4},x(5)},

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

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

det(isom):= -1$

ZZ(1):=diay(4)$

ZZ(2):=diay(2)$

ZZ(3):=diay(5)$

ZZ(4):=diay(1)$

ZZ(5):=diay(3)$

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

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

The isomorphism from g_{7,4.3} to gtildedelta$

was constructed in 2 steps and is given by$

the product matrix P*isom:=
mat((0,0,0,1,0,0,0),(0,1,0,0,0,0,0),(0,0,0,0,1,0,0),(1,0,0,0,0,0,0),(0,0,1,0,0,0
,0),(0,0,0,0,0,-1,0),(0,0,0,0,0,0,1))$

which we record here under the name PSI$