On products of two involutions in the orthogonal group of a vector space (Q1091473)

Let V be a vector space over a commutative field K such that the characteristic of K is distinct from 2. Assume that V is endowed with a symmetric nondegenerate bilinear form. For each \(\pi\) in the orthogonal group O(V) the subspace \(B(\pi)=image(\pi -1)\) is called the path of \(\pi\). The authors show: (A) If \(\rho\) and \(\sigma\) are involutions in O(V) and \(\pi =\rho \sigma\), then there is an orthogonal decomposition of V into orthogonally indecomposable \(\pi\)-modules that are invariant under \(\rho\) and \(\sigma\) ; (B) If \(\pi\in O(V)\), then there are involutions \(\rho\),\(\sigma\in O(V)\) such that \(\pi =\rho \sigma\) and \(B(\pi)=B(\rho)+B(\sigma)\) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable \(\pi\)-modules does not contain a term whose minimal polynomial is \((x-1)^{\alpha}\) where \(\alpha\) is even.