Bireflectionality in special orthogonal groups

TL;DR

This paper characterizes when elements of special orthogonal groups are bireflectional, linking this property to reversibility and providing conditions based on the dimension and structure of the quadratic space.

Contribution

It establishes a complete characterization of bireflectional elements in SO(V), connecting their properties to reversibility and the dimension of the space.

Findings

An element in SO(V) is bireflectional iff it is reversible.

All elements are bireflectional if dim V is odd or divisible by 4.

Special cases include hyperbolic planes over GF(2) or GF(3).

Abstract

It is shown that a transformation in the special orthogonal group SO(V) of a nondefective quadratic space over a field K is bireflectional (product of 2 involutions) if and only if it is reversible (conjugate to its inverse). Furthermore, all elements of SO(V) are bireflectional if and only if dim V is odd or divisible by 4, or V is a hyperbolic plane over GF(2) or GF(3).