  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.
