Bireflections of the commutator subgroup of an orthogonal group over the reals

Let $O(p,q)$ be the orthogonal groups of signature $(p,q)$ over the reals. It is shown that an element of the commutator subgroup $O(p,q{)}^{\mathrm{\prime }}$ of $O(p,q)$ is bireflectional (product of 2 involutions in $O(p,q{)}^{\mathrm{\prime }}$) if and only if it is reversible (conjugate to its inverse). Moreover, the bireflectional elements of $O(p,q{)}^{\mathrm{\prime }}$ are classified.