  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 algebraic
formulation 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 will
show, 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 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 P^e \simeq P\otimes P^T 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.
