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

Class XIII: (2 2 2 )

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, 1}, {0, 0, 0, 0, 0,  lam3}}));

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, 0, eps3}, {0, 0, 0, 0, eps3, 0}}));

eps1,2,3 have the same block size, so we may assume that e1 <= eps2 <= eps3:
Otherwise all configurations are possible:

Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1,eps3->-1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1,eps3->1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->1,eps3->1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->1,eps3->1}]

(* See ClassXIII_Solve.nb *)
S=(({{0, 1, 0, 0, 0, 0}, {0, 0, eps2, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {eps1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, eps3, 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/ClassXIII.pdf"]