Algebras with three anticommuting elements. I. Spinors and quaternions

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Abstract
A general construction of alternative algebras with three anticommuting elements and a unit is given. As an exhaustive result over the real and complex fields, we obtain the Clifford algebras H (quaternions), N1 (dihedral Clifford algebra which is related to real 2-spinors), and S1 (algebra of Pauli matrices which is related to complex 2-spinors). What is important is that the algebras N1 and S1 possess inverses everywhere except on a region akin to the light cone of the Minkowski space, while the quaternion algebra H has inverses everywhere except at the zero element. We discuss the reasons why the three algebras N1, H, and S1 are so difficult to distinguish in the representation space of 2×2 complex matrices.

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