Daviau showed the equivalence of matrix Dirac theory, formulated within a spinor bundle (S_x C _x^4 ), to a Clifford algebraic formulation within space Clifford algebra (C ( R ^3 , ) M _ 2 ( C ) P ) Pauli algebra (matrices) ≃ ℍ ⨁ ℍ ≃ biquaternions. We will show, that Daviau's map θ: ( : C ^4 M _ 2 ( C ) ) is an isomorphism. It is shown that Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation, which uses outer automorphisms. The connection between 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^ P 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.
A historical review of spinors is given together with a construction of spinor spaces as minimal left ideals of Clifford algebras. Spinor spaces of euclidean spaces over reals have a natural linear structure over reals, complex numbers or quaternions. Clifford algebras have involutions which induce bilinear forms or scalar products on spinor spaces. The automorphism groups of these scalar products of spinors are determined and also classified.