  In this paper we introduce the Dirac and spin-Dirac operators associated to a
connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac
operators associated with a Levi-Civita connection on a Riemannian (Lorentzian)
space(time) and calculate the square of these operators, which play an
important role in several topics of modern Mathematics, in particular in the
study of the geometry of moduli spaces of a class of black holes, the geometry
of NS-5 brane solutions of type II supergravity theories and BPS solitons in
some string theories. We obtain a generalized Lichnerowicz formula,
decompositions of the Dirac and spin-Dirac operators and their squares in terms
of the standard Dirac and spin-Dirac operators and using the fact that spinor
fields (sections of a spin-Clifford bundle) have representatives in the
Clifford bundle we present also a noticeable relation involving the spin-Dirac
and the Dirac operators.
