Products of involutions in \(O^ +(V)\) (Q1091474)

Let V be a finite-dimensional vector space over a commutative field K whose characteristic is distinct from 2. Assume V is endowed with a nondegenerate symmetric bilinear form. Let \(O^+(V)=\{\pi \in O(V):\) det \(\pi\) \(=1\}\). The authors show: An element \(\pi \in O^+(V)\) is a product of two involutions if and only if dim \(V\not\equiv 2 mod 4\) or an orthogonal decomposition of V into orthogonally indecomposable \(\pi\)- modules contains a \(\pi\)-module of odd dimension.