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

Class IX: (42)

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, 1, 0, 0, 0}, {0, 0, lam1, 1, 0, 0}, {0, 0, 0, lam1, 0, 0}, {0, 0, 0, 0,  lam2, 1}, {0, 0, 0, 0, 0,  lam2}}));

Eigenvalues[V]

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

All sign configurations are possible

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

S=(({{0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {eps1, 0, 0, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {0, 0, 0, 0, eps2, 0}}));

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/ClassIX.pdf"]