The invariant decomposition is a decomposition of the elements of \text{Pin}(p,q,r) groups 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 readsEvery k-reflection can be decomposed into \lceil k/2 \rceil commuting factors.
