Products of two involutions in orthogonal and symplectic groups

Abstract: An element of a group is called bireflectional when it is the product of two involutions of the group (i.e. elements of order 1 or 2). If an element is bireflectional then it is conjugated to its inverse. It is known that all elements of orthogonal groups of quadratic forms are bireflectional. F. B\"unger has characterized the elements of unitary groups (over fields of characteristic not $2$) that are bireflectional. Yet in symplectic groups over fields with characteristic different from 2, in general there are elements that are conjugated to their inverse but are not bireflectional (however, over fields of characteristic 2, every element of a symplectic group is bireflectional). In this article, we characterize the bireflectional elements of symplectic groups in terms of Wall invariants, over fields of characteristic not 2: the result is cited without proof in B\"unger's PhD thesis, and attributed to Klaus Nielsen. We also take advantage of our approach to give a simplified proof of Wonenburger's corresponding result for orthogonal groups and general linear groups.