  We investigate using Clifford algebra methods the theory of algebraic dotted
and undotted spinor fields over a Lorentzian spacetime and their realizations
as matrix spinor fields, which are the usual dotted and undotted two component
spinor fields. We found that some ad hoc rules postulated for the covariant
derivatives of Pauli sigma matrices and also for the Dirac gamma matrices in
General Relativity cover important physical meaning, which is not apparent in
the usual matrix presentation of the theory of two components dotted and
undotted spinor fields. We also discuss some issues related to the the previous
one and which appear in a proposed "unified" theory of gravitation and
electromagnetism which use two components dotted and undotted spinor fields and
also paravector fields, which are particular sections of the even subundle of
the Clifford bundle of spacetime.
