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

Class III: (3bar(3))

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=(({{ sigma1, tau1, 1, 0, 0, 0}, {-tau1, sigma1, 0, 1, 0, 0}, {0, 0, sigma1, tau1, 1, 0}, {0, 0, -tau1, sigma1, 0, 1}, {0, 0, 0, 0,  sigma1, tau1}, {0, 0, 0, 0, -tau1,  sigma1}}));

Eigenvalues[V]

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

Eigenvalues[W]

(* Found with FindSPermutations.m: permutation 6,6 with leading B *)
S=(({{0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 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/ClassIII.pdf"]