Invariant decomposition

Summary

The invariant decomposition is a decomposition of the elements of pin groups $\text{Pin}(p,q,r)$ into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of $k$ oriented reflections, the invariant decomposition theorem reads

Every $k$-reflection can be decomposed into $\lceil k/2 \rceil$ commuting factors.

It is named the invariant decomposition because these factors are the invariants of the $k$-reflection $R \in \text{Pin}(p,q,r)$. A well known special case is the Chasles' theorem, which states that any rigid body motion in $\text{SE}(3)$ can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Every quadreflection can be decomposed into commuting bireflections". In this form the statement is also valid for e.g. the spacetime algebra $\text{SO}(3,1)$, where any Lorentz transformation can be decomposed into a commuting rotation and boost.