  In this paper we present a new, elementary derivation of non-relativistic
spin using exclusively real algebraic methods. To do this, we formulate a novel
method to decompose the domain of a real endomorphism according to its
algebraic properties. We reveal non-commutative multipole tensors as the
primary physically meaningful observables of spin, and indicate that spin is
fundamentally geometric in nature. In so doing, we demonstrate that neither
dynamics nor complex numbers are essential to the fundamental description of
spin.
