The special orthogonal group is trireflectional

Abstract

Let K be a field of even characteristic, V a finite-dimensional vector space over K, and SO(V ) the special orthogonal group. Then SO(V ) is trireflectional, provided dim V > 2 and SO(V ) = O + (4, 2). Problem statement.Let V be a finite-dimensional vector space over a field K. A mapping q : V −→ K is called a quadratic form if q(αx) = α 2 q(x) for all α ∈ K and all x ∈ V and if there is a symmetric bilinear form b q : V × V −→ K such that b q (x, y) := q(x+y)−q(x)−q(y) for x, y ∈ V . We call the bilinear form b q nondegenerate or nonsingular if b q (x, y) = 0 for all y ∈ V implies x = 0. The orthogonal group of q is defined by