On the equivalence of Daviau's space Clifford algebraic, Hestenes' and Parra's formulations of (real) Dirac theory

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Recently Daviau showed the equivalence of ordinary matrix based Dirac theory -formulated within a spinor bundle Sₓ ≃ C⁴ₓ-, to a Clifford algebraic formulation within space Clifford algebra CL(R³,delta) ≃ M₂(C) ≃ P ≃ Pauli algebra (matrices) ≃ H H ≃ biquaternions. We will show, that Daviau's map theta: C⁴ M₂(C) is an isomorphism. Furthermore it is shown that Hestenes' and Parra's formulations are equivalent to Daviau's space Clifford algebra formulation, which however uses outer automorphisms. The connection between such different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra Pᵉ ≃ P⊗ Pᵗ on P. The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau and others to interprete the right action as the iso-spin freedom.