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.



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