A note on bireflectional elements of an algebra

Abstract

A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field $\mathbb{F}$ is algebraically closed with characteristic other than $2$. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over $\mathbb{F}$.