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  "body": "\n## Summary\n\nThe **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|0TKYAEJ8N7]], pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of $k$ oriented reflections, the invariant decomposition theorem reads\n\n\u003e Every $k$-reflection can be decomposed into $\\lceil k/2 \\rceil$ commuting factors.\n\nIt 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.\n",
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