Proof of Proposition 3.5 (i): We compute spectrum of 4x4 matrix H

(* Define metric (and inverse) *)
GG = DiagonalMatrix[{s1,s2,s3,s4}];
GG//MatrixForm

Ginv = Inverse[GG];
Ginv//MatrixForm

(* define covector *)
xi = {xi0,xi1,xi2,xi3};

(* define H matrix *)
HH = (xi.Ginv.xi) Ginv - Table[ (xi.Ginv)[[i]] (xi.Ginv)[[j]],{i,1,4},{j,1,4}];
HH//MatrixForm

Signature (+,+,+,+)

Eigenvalues[HH/.{s1->1,s2->1,s3->1,s4->1}]

Signature (-,+,+,+)

Eigenvalues[HH/.{s1->-1,s2->1,s3->1,s4->1}]

Signature (-,-,+,+)

Eigenvalues[HH/.{s1->-1,s2->-1,s3->1,s4->1}]

We do not need to consider the remaining signatures since matrix H is invariant under g -> -g.