Bireflectionality in the commutator subgroup of a finite orthogonal group

We classify the bireflections (products of 2 involutions) in the commutator subgroup G an orthogonal group O(V) over a finite field GF(q) of characteristic not 2. We show that every element of G is a bireflection if it is reversible (conjugate to its inverse in G), except when $q\equiv 3\mathrm{m}\mathrm{o}\mathrm{d}4,\dim V\equiv 2\mathrm{m}\mathrm{o}\mathrm{d}4$ and $V$ is hyperbolic. We also classify the reversible elements of G.