Products of Involutions in O + (V)

AbstractLet V be a finite-dimensional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. The special orthogonal group is O+(V):={πϵO(V): det π = 1}. Main result: An element π ∈ O+ (V) is a product of two involutions in O+ (V) ifand only if dim V ≢ 2 mod 4 or an orthogonal decomposition of V into orthogonally indecomposable π-modules contains a π-module of odd dimension