  The symmetries described by Pin groups are the result of combining a finite
number of discrete reflections in (hyper)planes. The current work shows how an
analysis using geometric algebra provides a picture complementary to that of
the classic matrix Lie algebra approach, while retaining information about the
number of reflections in a given transformation. This imposes a graded
structure on Lie groups, which is not evident in their matrix representation.
By embracing this graded structure, the invariant decomposition theorem was
proven: any composition of $k$ linearly independent reflections can be
decomposed into $\lceil k/2 \rceil$ commuting factors, each of which is the
product of at most two reflections. This generalizes a conjecture by M. Riesz,
and has e.g. the Mozzi-Chasles' theorem as its 3D Euclidean special case. To
demonstrate its utility, we briefly discuss various examples such as Lorentz
transformations, Wigner rotations, and screw transformations. The invariant
decomposition also directly leads to closed form formulas for the exponential
and logarithmic function for all Spin groups, and identifies element of
geometry such as planes, lines, points, as the invariants of $k$-reflections.
We conclude by presenting novel matrix/vector representations for geometric
algebras $\mathbb{R}_{pqr}$, and use this in E(3) to illustrate the
relationship with the classic covariant, contravariant and adjoint
representations for the transformation of points, planes and lines.
