On products of two involutions in the orthogonal group of a vector space

Abstract

Let V be a finite-dimentional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. Results: (A) Let π = ρσ, where π, ρ, σ ∈ O(V) and ρ, σ are involutions. There exists an orthogonal decomposition of V into orthogonally indecomposable π-modules which are simultaneously invariant under ρ and σ. (B) Let π ∈ O(V).One can find involutions ρ, σ ∈ O(V) such that π = ρσ and B(π) = B(ρ) + B(σ) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable π-modules does not contain a term whose minimum polynomial is (x-1)α where α is even. © 1987.