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Recently Daviau showed the equivalence of ordinary matrix based Dirac theory-formulated within a spinor bundle S_x \simeq C^4_x-, to a Clifford algebraicformulation within space Clifford algebra CL(R^3,delta) \simeq M_2(C) \simeq P\simeq Pauli algebra (matrices) \simeq H \oplu H \simeq biquaternions. We willshow, that Daviau's map theta : C^4 \mapsto M_2(C) is an isomorphism.Furthermore it is shown that Hestenes' and Parra's formulations are equivalentto Daviau's space Clifford algebra formulation, which however uses outerautomorphisms. The connection between such different formulations is quiteremarkable, since it connects the left and right action on the Pauli algebraitself viewed as a bi-module with the left (resp. right) action of theenveloping algebra P^e \simeq P\otimes P^T on P. The isomorphism established inthis article and given by Daviau's map does clearly show that right and leftactions 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.Comment: 10 pages, nothing special, Submitted to IJTP special issue about the  Ixtapa conference on `Clifford algebras and their applications in  mathematical physics' eds. D. Ritz Finkelstein, Z. Oziewic

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