SetDirectory["/www/user/fdahl/papers/Conjugation/"];
<<kappaLib.m
<<Petrov.m

Class XV: (2 2 11 )

B=(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}}));

V=(({{ lam1, 1, 0, 0, 0, 0}, {0, lam1, 0, 0, 0, 0}, {0, 0, lam2, 1, 0, 0}, {0, 0, 0, lam2, 0, 0}, {0, 0, 0, 0,  lam3, 0}, {0, 0, 0, 0, 0,  lam4}}));

Eigenvalues[V]

W=(({{0, eps1, 0, 0, 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, 0, eps2, 0, 0}, {0, 0, eps2, 0, 0, 0}, {0, 0, 0, 0, eps3, 0}, {0, 0, 0, 0, 0, eps4}}));

Eigenvalues[W]

eps3 and eps4 have same block size we we may assume that eps3 <= eps4.
By above expression for Eigenvalues[W], we see that eps3<eps4.

W=W/.{eps3->-1, eps4->1};

Eigenvalues[W]

Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->1}]

Thus: eps3=-1, eps4=+1 and eps1,eps2 are arbitrary, but with eps1<=eps2

S=(({{0, 0, 0, 1, 0, 0}, {1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, -(1/(Sqrt[2])), (1/(Sqrt[2]))}, {0, 0, eps2, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {0, 0, 0, 0, (1/(Sqrt[2])), (1/(Sqrt[2]))}}));

Check that S is in set mathcal(S)

Transpose[S].B.S==W

Compute result

res=S.V.Inverse[S];
res//MatrixForm

Petrov[res]

Export notebook as .pdf

NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/ClassXV.pdf"]