generic derivation : delta:= mat((xi(1,1),xi(1,2),0,0,0,0), (xi(2,1),xi(2,2),0,0,0,0), (xi(3,1),xi(3,2),xi(3,3), - xi(1,2),0,0), (xi(4,1),xi(4,2), - xi(2,1),xi(3,3) + xi(2,2) - xi(1,1),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),xi(4,2) - xi(3,1),xi(3,3) + xi(2,2))) The nonzero adjoints are delta(6,1), delta(6,2), and delta(5,2)+delta(6,4), -delta(5,1)+delta(6,3) Hence -delta(5,2) and delta(6,4) are adjoint related and so are delta(5,1) and delta(6,3) as well. [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 1 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 1 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 -1 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 0 0 0 0 0] [ ] [-1 0 0 0 0 0] delta is nilpotent if and only if xi(2,2)=-xi(1,1) , xi(3,3)=xi(1,1) and finally the matrix A:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent generic nilpotent derivation : delta:= [xi(1,1) xi(1,2) 0 0 0 0] [ ] [xi(2,1) - xi(1,1) 0 0 0 0] [ ] [xi(3,1) xi(3,2) xi(1,1) - xi(1,2) 0 0] [ ] [xi(4,1) xi(4,2) - xi(2,1) - xi(1,1) 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) xi(4,2) - xi(3,1) 0] with A:= [xi(1,1) xi(1,2) ] [ ] [xi(2,1) - xi(1,1)] nilpotent The generic automorphism of g_{6,1} as in (art ijac1 section6) : phi:= mat((b(1,1),b(1,2),0,0,0,0), (b(2,1),b(2,2),0,0,0,0), (b(3,1),b(3,2),b(1,1)*u, - b(1,2)*u,0,0), (b(4,1),b(4,2), - b(2,1)*u,b(2,2)*u,0,0), (b(5,1),b(5,2),b(5,3),b(5,4),b(2,2)*b(1,1) - b(2,1)*b(1,2),0), (b(6,1),b(6,2),b(6,3),b(6,4), b(4,2)*b(1,1) - b(4,1)*b(1,2) + b(3,2)*b(2,1) - b(3,1)*b(2,2), u*(b(2,2)*b(1,1) - b(2,1)*b(1,2)))) 4 3 det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2)) *u We consider here in this case 2 the case where A = 0. A:= [0 0] [ ] [0 0] generic nilpotent derivation in the case 1 : delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [xi(4,1) xi(4,2) 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) xi(4,2) - xi(3,1) 0] by subtracting adjoints one then may suppose xi(5,1)=xi(5,2)=xi(6,1)=xi(6,2)=0 phi:= mat((b(1,1),b(1,2),0,0,0,0), (b(2,1),b(2,2),0,0,0,0), (b(3,1),b(3,2),b(1,1)*u, - b(1,2)*u,0,0), (b(4,1),b(4,2), - b(2,1)*u,b(2,2)*u,0,0), (b(5,1),b(5,2),b(5,3),b(5,4),b(2,2)*b(1,1) - b(2,1)*b(1,2),0), (b(6,1),b(6,2),b(6,3),b(6,4), b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1), (b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)) 4 3 det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2)) *u delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [xi(4,1) xi(4,2) 0 0 0 0] [ ] [ 0 0 xi(5,3) xi(5,4) 0 0] [ ] [ 0 0 xi(6,3) xi(6,4) xi(4,2) - xi(3,1) 0] We denote this delta by the shortform shortformdelta:={xi(3,1), xi(3,2), ss, xi(4,1), xi(4,2), ss, xi(5,3), xi(5,4), ss, xi(6,3), xi(6,4)} paramindexeslist:={{3,1},{3,2},{4,1},{4,2},{5,3},{5,4},{6,3},{6,4}} shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (b(2,2)*xi(5,3) + b(2,1)*xi(5,4))/u, (b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u, ss, ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(2,2)*xi(5,3) + b(2,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2 ,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)))*u + ((b(3,2)*b(2, 1) - b(3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)*b(2,1)*b(1,1))*xi(5,4 ) + ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1, 2))*xi(5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2, 1)*xi(3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u), (((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,1) - b(4,1)*b(1,2)*b(1,1) + b(4,2)*b(1,1) **2)*xi(5,4) + ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1 ,1)*xi(3,2))*b(5,3))*u + ((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,2) - b(4,1)*b(1,2) **2 + b(4,2)*b(1,2)*b(1,1))*xi(5,3) + (b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b( 1,2) + b(4,2)*b(1,1))*(b(1,2)*xi(5,3) + b(1,1)*xi(5,4)) - ((b(5,4)*b(1,1) + b(5, 3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=(b(2,2)*xi(5,3) + b(2,1)*xi(5,4))/u$ deltaprimemodg(5,4):=(b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u$ deltaprimemodg(6,3):=((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b( 1,1))*(b(2,2)*xi(5,3) + b(2,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4 ,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4 )))*u + ((b(3,2)*b(2,1) - b(3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)* b(2,1)*b(1,1))*xi(5,4) + ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1,2))*xi(5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + ( b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)$ deltaprimemodg(6,4):=(((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,1) - b(4,1)*b(1,2)*b( 1,1) + b(4,2)*b(1,1)**2)*xi(5,4) + ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(5,4) + ( b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))*u + ((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b( 2,2) - b(4,1)*b(1,2)**2 + b(4,2)*b(1,2)*b(1,1))*xi(5,3) + (b(3,2)*b(2,1) - b(3,1 )*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(1,2)*xi(5,3) + b(1,1)*xi(5,4)) - (( b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1 ,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,(b(2,2)* xi(5,3) + b(2,1)*xi(5,4))/u,(b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u,0,0),(0,0,((b(3, 2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(2,2)*xi(5,3) + b(2 ,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b( 1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)))*u + ((b(3,2)*b(2,1) - b (3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)*b(2,1)*b(1,1))*xi(5,4) + (( b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1,2))*xi (5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2,1)*xi( 3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u),(((b(3,2)*b(1,1) - b(3,1)*b (1,2))*b(2,1) - b(4,1)*b(1,2)*b(1,1) + b(4,2)*b(1,1)**2)*xi(5,4) + ((b(1,2)*xi(4 ,1) - b(1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))*u + ((b (3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,2) - b(4,1)*b(1,2)**2 + b(4,2)*b(1,2)*b(1,1))* xi(5,3) + (b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(1,2 )*xi(5,3) + b(1,1)*xi(5,4)) - ((b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1 )) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2 ,2)*b(1,1) - b(2,1)*b(1,2))*u),(xi(4,2) - xi(3,1))*u,0))$ ********* Suppose that xi(5,3) and xi(5,4) are both zero.$ xi(5,3):=0$ xi(5,4):=0$ Take first:$ b(5,3):=0$ b(5,4):=0$ shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, 0, 0, ss, b(2,2)*xi(6,3) + b(2,1)*xi(6,4), b(1,2)*xi(6,3) + b(1,1)*xi(6,4)}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=b(2,2)*xi(6,3) + b(2,1)*xi(6,4)$ deltaprimemodg(6,4):=b(1,2)*xi(6,3) + b(1,1)*xi(6,4)$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0), (0,0,b(2,2)*xi(6,3) + b(2,1)*xi(6,4),b(1,2)*xi(6,3) + b(1,1)*xi(6,4),(xi(4,2) - xi(3,1))*u,0))$ ******Suppose that xi(6,3) and xi(6,4) are both zero:$ xi(6,3):=0$ xi(6,4):=0$ shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, 0, 0, ss, 0, 0}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0), (0,0,0,0,(xi(4,2) - xi(3,1))*u,0))$ If we take b(1,1):=0,b(1,2):=1,b(2,1):=1,b(2,2):=0 we get:$ shortformdeltaprimemodadg:={ - xi(4,2)*u, - xi(4,1)*u, ss, - xi(3,2)*u, - xi(3,1)*u, ss, 0, 0, ss, 0, 0}$ deltaprimemodg(3,1):= - xi(4,2)*u$ deltaprimemodg(3,2):= - xi(4,1)*u$ deltaprimemodg(4,1):= - xi(3,2)*u$ deltaprimemodg(4,2):= - xi(3,1)*u$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(phi):=u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),( - xi(4,2)*u, - xi(4,1)*u,0,0,0,0),( - xi(3,2)* u, - xi(3,1)*u,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,(xi(4,2) - xi(3,1))*u,0))$ Hence (xi(3,1),xi(3,2)) and (xi(4,1),xi(4,2)) are interchanged$ !Now,! as! delta! neq! 0,! ! we! have! not! simultaneously! (xi(3,1),xi(3,2))! a nd (xi(4,1),xi(4,2))! ! both! zero.$ Hence we may suppose that xi(3,1),xi(3,2) are not both zero.$ clear b(1,1),b(1,2),b(2,1),b(2,2)$$ shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, 0, 0, ss, 0, 0}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0), (0,0,0,0,(xi(4,2) - xi(3,1))*u,0))$ ****** Suppose that xi(3,2) = 0.$ xi(3,2):=0$ Then xi(3,1) neq 0 and one can suppose xi(3,1):=1.$ xi(3,1):=1$ shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - b(2, 2)*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2) - b(1,1))*b(1,2)*u)/(b(2,2)*b(1,1) - b(2,1)*b( 1,2)), ss, ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) - b(2,1))*b(2,2)*u)/(b(2,2)*b(1,1) - b(2,1)*b( 1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - b(2,1)*b(1,2))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, 0, 0, ss, 0, 0}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - b(2,2)*b(1, 1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=((b(1,2)*xi(4,1) - b(1,1)*xi(4,2) - b(1,1))*b(1,2)*u)/(b(2, 2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) - b(2,1))*b(2,2)*u)/(b(2, 2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - b(2,1)*b(1, 2))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - b(2,2)*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),((b(1,2)*xi(4,1) - b(1,1)*xi( 4,2) - b(1,1))*b(1,2)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(((b(2,2)*xi(4 ,1) - b(2,1)*xi(4,2) - b(2,1))*b(2,2)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),( - ((b (1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - b(2,1)*b(1,2))*u)/(b(2,2)*b(1,1) - b(2, 1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,(xi(4,2) - 1)*u,0))$ We take :$ b(1,2):=0$ Then we get:$ Then one gets deltaprime(3,1)=k by taking$ u:=k$ shortformdeltaprimemodadg:={k, 0, ss, ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) - b(2,1))*k)/b(1,1), xi(4,2)*k, ss, 0, 0, ss, 0, 0}$ deltaprimemodg(3,1):=k$ deltaprimemodg(3,2):=0$ deltaprimemodg(4,1):=((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) - b(2,1))*k)/b(1,1)$ deltaprimemodg(4,2):=xi(4,2)*k$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=0$ deltaprimemodg(6,4):=0$ det(phi):=b(2,2)**4*b(1,1)**4*k**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(k,0,0,0,0,0),(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) - b(2,1))*k)/b(1,1),xi(4,2)*k,0,0,0,0),(0,0,0,0,0,0),(0,0,0,0,(xi(4,2) - 1)*k,0 ))$ shortformdeltaprime:=shortformdeltaprime$ Thus if xi(3,2) = 0 we are reduced to:$ shortformdeltaprime ={1,0,SS,epsilon,a,SS,0,0,SS,0,0}$ where a =xi(4,2) is any complex and epsilon = xi(4,1) = 0,1.$