The generic automorphism phi of g_{6,n2} as computed by calculautom6_n2.red : They fall into 2 kinds. First kind : phi:= mat((b(1,1),b(1,2),0,0,0,0), (b(2,1),b(2,2),0,0,0,0), (0,0,b(3,3),b(3,4),0,0), (0,0,b(4,3),b(4,4),0,0), (b(5,1),b(5,2),b(5,3),b(5,4), - b(2,1)*b(1,2) + b(2,2)*b(1,1),0), (b(6,1),b(6,2),b(6,3),b(6,4),0, - b(4,3)*b(3,4) + b(4,4)*b(3,3))) 2 det(phi):=( - b(4,3)*b(3,4) + b(4,4)*b(3,3)) 2 *( - b(2,1)*b(1,2) + b(2,2)*b(1,1)) Second kind : psi:= mat((0,0,b(1,3),b(1,4),0,0), (0,0,b(2,3),b(2,4),0,0), (b(3,1),b(3,2),0,0,0,0), (b(4,1),b(4,2),0,0,0,0), (b(5,1),b(5,2),b(5,3),b(5,4),0, - b(2,3)*b(1,4) + b(2,4)*b(1,3)), (b(6,1),b(6,2),b(6,3),b(6,4), - b(4,1)*b(3,2) + b(4,2)*b(3,1),0)) 2 det(psi):= - ( - b(4,1)*b(3,2) + b(4,2)*b(3,1)) 2 *( - b(2,3)*b(1,4) + b(2,4)*b(1,3)) generic derivation : delta:= [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(1,1) + xi(2,2) 0 ] [ ] [xi(6,1) xi(6,2) xi(6,3) xi(6,4) 0 xi(3,3) + xi(4,4)] [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 3. Then A:=((0,1),(0,0)); B:=((0,1),(0,0)) xi(1,1):=0 xi(1,2):=1 xi(2,1):=0 xi(2,2):=0 xi(3,3):=xi(3,3) xi(3,4):=xi(3,4) xi(4,3):=xi(4,3) xi(4,4):=xi(4,4) delta:= [ 0 1 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 1 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 1 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 1 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}} With the first kind automorphism one gets$ shortformdeltaprimemodadg:={( - b(5,3)*b(4,3) + ( - b(4,3)*xi(5,4) + b(4,4)*xi(5 ,3))*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3)), ( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3))*( - b(2,1)*b(1,2) + b(2,2)*b(1,1))))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3)), ss, ( - b(6,1)*b(2,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3))*( - b(2,1)*xi(6,2) + b(2, 2)*xi(6,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)), ( - ( - b(6,1)*b(1,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3))*( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))}$ deltaprimemodg(5,3):=( - b(5,3)*b(4,3) + ( - b(4,3)*xi(5,4) + b(4,4)*xi(5,3))*( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3))$ deltaprimemodg(5,4):=( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3) )*( - b(2,1)*b(1,2) + b(2,2)*b(1,1))))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3))$ deltaprimemodg(6,1):=( - b(6,1)*b(2,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3))*( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1)))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ deltaprimemodg(6,2):=( - ( - b(6,1)*b(1,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3))* ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1))$ det(AUTOM):=( - b(4,3)*b(3,4) + b(4,4)*b(3,3))**2*( - b(2,1)*b(1,2) + b(2,2)*b(1 ,1))**2$ DELTAPRIMEMODADG:= 2 - b(2,1)*b(1,1) b(1,1) mat((----------------------------------,----------------------------------,0,0,0 - b(2,1)*b(1,2) + b(2,2)*b(1,1) - b(2,1)*b(1,2) + b(2,2)*b(1,1) ,0), 2 - b(2,1) b(2,1)*b(1,1) (----------------------------------,----------------------------------,0,0,0 - b(2,1)*b(1,2) + b(2,2)*b(1,1) - b(2,1)*b(1,2) + b(2,2)*b(1,1) ,0), 2 - b(4,3)*b(3,3) b(3,3) (0,0,----------------------------------,----------------------------------,0 - b(4,3)*b(3,4) + b(4,4)*b(3,3) - b(4,3)*b(3,4) + b(4,4)*b(3,3) ,0), 2 - b(4,3) b(4,3)*b(3,3) (0,0,----------------------------------,----------------------------------,0 - b(4,3)*b(3,4) + b(4,4)*b(3,3) - b(4,3)*b(3,4) + b(4,4)*b(3,3) ,0), (0,0,( - b(5,3)*b(4,3) + ( - b(4,3)*xi(5,4) + b(4,4)*xi(5,3)) *( - b(2,1)*b(1,2) + b(2,2)*b(1,1)))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3) ),( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3)) *( - b(2,1)*b(1,2) + b(2,2)*b(1,1))))/( - b(4,3)*b(3,4) + b(4,4)*b(3,3)),0,0), (( - b(6,1)*b(2,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3))*( - b(2,1)*xi(6,2) + b(2,2)*xi(6,1)) )/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),( - ( - b(6,1)*b(1,1) + ( - b(4,3)*b(3,4) + b(4,4)*b(3,3)) *( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))))/( - b(2,1)*b(1,2) + b(2,2)*b(1,1)),0,0,0,0)) With the second kind automorphism one gets$ shortformdeltaprimemodadg:={( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6 ,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)), ( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)), ss, ( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2, 4)*xi(5,3)))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)), ( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))}$ deltaprimemodg(5,3):=( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1))$ deltaprimemodg(5,4):=( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1) )*( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1))$ deltaprimemodg(6,1):=( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2,4)*xi(5,3)))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))$ deltaprimemodg(6,2):=( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))* ( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))$ det(AUTOM):= - ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))**2*( - b(2,3)*b(1,4) + b(2,4)* b(1,3))**2$ DELTAPRIMEMODADG:= 2 - b(2,3)*b(1,3) b(1,3) mat((----------------------------------,----------------------------------,0,0,0 - b(2,3)*b(1,4) + b(2,4)*b(1,3) - b(2,3)*b(1,4) + b(2,4)*b(1,3) ,0), 2 - b(2,3) b(2,3)*b(1,3) (----------------------------------,----------------------------------,0,0,0 - b(2,3)*b(1,4) + b(2,4)*b(1,3) - b(2,3)*b(1,4) + b(2,4)*b(1,3) ,0), 2 - b(4,1)*b(3,1) b(3,1) (0,0,----------------------------------,----------------------------------,0 - b(4,1)*b(3,2) + b(4,2)*b(3,1) - b(4,1)*b(3,2) + b(4,2)*b(3,1) ,0), 2 - b(4,1) b(4,1)*b(3,1) (0,0,----------------------------------,----------------------------------,0 - b(4,1)*b(3,2) + b(4,2)*b(3,1) - b(4,1)*b(3,2) + b(4,2)*b(3,1) ,0), (0,0,( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6,1)) *( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1) ),( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1)) *( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)),0,0), (( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2,4)*xi(5,3)) )/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)),( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1)) *( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)),0,0,0,0)) To keep deltaprime(1,1):=0 and deltaprime(1,2):=1 under the first kind,$ one has to take :$ b(2,2):=b(1,1)$ b(2,1):=0$ To keep deltaprime(3,3):=0 and deltaprime(3,4):=1 under the first kind,$ one has to take :$ b(4,3):=0$ b(4,4):=b(3,3)$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={(b(1,1)**2*xi(5,3))/b(3,3), ( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3))*b(1,1)**2))/b(3,3) **2, ss, (b(3,3)**2*xi(6,1))/b(1,1), ( - ( - b(6,1)*b(1,1) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(3,3)**2))/b(1,1) **2}$ deltaprimemodg(5,3):=(b(1,1)**2*xi(5,3))/b(3,3)$ deltaprimemodg(5,4):=( - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3) )*b(1,1)**2))/b(3,3)**2$ deltaprimemodg(6,1):=(b(3,3)**2*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=( - ( - b(6,1)*b(1,1) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1) )*b(3,3)**2))/b(1,1)**2$ det(AUTOM):=b(3,3)**4*b(1,1)**4$ DELTAPRIMEMODADG:= mat((0,1,0,0,0,0), (0,0,0,0,0,0), (0,0,0,1,0,0), (0,0,0,0,0,0), 2 b(1,1) *xi(5,3) (0,0,-----------------, b(3,3) 2 - ( - b(5,3)*b(3,3) + ( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3))*b(1,1) ) ----------------------------------------------------------------------,0,0) 2 b(3,3) , 2 b(3,3) *xi(6,1) (-----------------, b(1,1) 2 - ( - b(6,1)*b(1,1) + ( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(3,3) ) ----------------------------------------------------------------------,0,0, 2 b(1,1) 0,0)) With the second kind automorphism one gets$ shortformdeltaprimemodadg:={( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6 ,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)), ( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)), ss, ( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2, 4)*xi(5,3)))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)), ( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))}$ deltaprimemodg(5,3):=( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6,1))*( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1))$ deltaprimemodg(5,4):=( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1) )*( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1))$ deltaprimemodg(6,1):=( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2,4)*xi(5,3)))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))$ deltaprimemodg(6,2):=( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))* ( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3))$ det(AUTOM):= - ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))**2*( - b(2,3)*b(1,4) + b(2,4)* b(1,3))**2$ DELTAPRIMEMODADG:= 2 - b(2,3)*b(1,3) b(1,3) mat((----------------------------------,----------------------------------,0,0,0 - b(2,3)*b(1,4) + b(2,4)*b(1,3) - b(2,3)*b(1,4) + b(2,4)*b(1,3) ,0), 2 - b(2,3) b(2,3)*b(1,3) (----------------------------------,----------------------------------,0,0,0 - b(2,3)*b(1,4) + b(2,4)*b(1,3) - b(2,3)*b(1,4) + b(2,4)*b(1,3) ,0), 2 - b(4,1)*b(3,1) b(3,1) (0,0,----------------------------------,----------------------------------,0 - b(4,1)*b(3,2) + b(4,2)*b(3,1) - b(4,1)*b(3,2) + b(4,2)*b(3,1) ,0), 2 - b(4,1) b(4,1)*b(3,1) (0,0,----------------------------------,----------------------------------,0 - b(4,1)*b(3,2) + b(4,2)*b(3,1) - b(4,1)*b(3,2) + b(4,2)*b(3,1) ,0), (0,0,( - b(5,1)*b(4,1) + ( - b(4,1)*xi(6,2) + b(4,2)*xi(6,1)) *( - b(2,3)*b(1,4) + b(2,4)*b(1,3)))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1) ),( - ( - b(5,1)*b(3,1) + ( - b(3,1)*xi(6,2) + b(3,2)*xi(6,1)) *( - b(2,3)*b(1,4) + b(2,4)*b(1,3))))/( - b(4,1)*b(3,2) + b(4,2)*b(3,1)),0,0), (( - b(6,3)*b(2,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1))*( - b(2,3)*xi(5,4) + b(2,4)*xi(5,3)) )/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)),( - ( - b(6,3)*b(1,3) + ( - b(4,1)*b(3,2) + b(4,2)*b(3,1)) *( - b(1,3)*xi(5,4) + b(1,4)*xi(5,3))))/( - b(2,3)*b(1,4) + b(2,4)*b(1,3)),0,0,0,0)) We get using the first kind deltaprime(5,4) and deltaprime(6,2):=0 if we take :$ b(5,3):=(( - b(3,3)*xi(5,4) + b(3,4)*xi(5,3))*b(1,1)**2)/b(3,3)$ b(6,1):=(( - b(1,1)*xi(6,2) + b(1,2)*xi(6,1))*b(3,3)**2)/b(1,1)$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={(b(1,1)**2*xi(5,3))/b(3,3), 0, ss, (b(3,3)**2*xi(6,1))/b(1,1), 0}$ deltaprimemodg(5,3):=(b(1,1)**2*xi(5,3))/b(3,3)$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,1):=(b(3,3)**2*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(3,3)**4*b(1,1)**4$ DELTAPRIMEMODADG:= [ 0 1 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 1 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(1,1) *xi(5,3) ] [ 0 0 ----------------- 0 0 0] [ b(3,3) ] [ ] [ 2 ] [ b(3,3) *xi(6,1) ] [----------------- 0 0 0 0 0] [ b(1,1) ] Hence we may suppose xi(6,2):=0$ xi(6,2):=0$ With the first kind automorphism one gets$ shortformdeltaprimemodadg:={(b(1,1)**2*xi(5,3))/b(3,3), 0, ss, (b(3,3)**2*xi(6,1))/b(1,1), 0}$ deltaprimemodg(5,3):=(b(1,1)**2*xi(5,3))/b(3,3)$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,1):=(b(3,3)**2*xi(6,1))/b(1,1)$ deltaprimemodg(6,2):=0$ det(AUTOM):=b(3,3)**4*b(1,1)**4$ DELTAPRIMEMODADG:= [ 0 1 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 1 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ b(1,1) *xi(5,3) ] [ 0 0 ----------------- 0 0 0] [ b(3,3) ] [ ] [ 2 ] [ b(3,3) *xi(6,1) ] [----------------- 0 0 0 0 0] [ b(1,1) ] Then, we clearly are reduced in this case 3 to:$ shortformdeltaprime ={eta,0,SS,epsilon,0}$ where epsilon=xi(6,1)=0,1 . and eta =xi(5,3):=0,1$