Bireflectionality of orthogonal and symplectic groups of characteristic 2

Abstract

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.