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

TL;DR

This paper characterizes and classifies bireflectional elements within the commutator subgroup of real orthogonal groups, linking their structure to the property of being reversible, and provides a complete classification.

Contribution

It establishes a precise criterion for bireflectionality in the commutator subgroup of real orthogonal groups and classifies all such elements.

Findings

Bireflectional elements are exactly those that are reversible.

A complete classification of bireflectional elements in $O(p,q)'$ is provided.

The paper links bireflectionality to conjugation properties within the group.

Abstract

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 $O(p,q)'$ of $O(p,q)$ is bireflectional (product of 2 involutions in $O(p,q)'$) if and only if it is reversible (conjugate to its inverse). Moreover, the bireflectional elements of $O(p, q)'$ are classified.