Generic intersection of orthogonal groups

Abstract

Let V be a finite-dimensional complex vector space and let g, h: V x V --> C be two non-degenerate symmetric bilinear forms. Let G, H be the groups of isometries of g, h, respectively. If the endomorphism L: V --> V associated to g, h is diagonalizable, then dim(G ∩ H) = Σ(i=1,...,r) (m(i) over 2), where m(i), i=1,...,r, are the dimensions of the eigenspaces of L.