  We take a Dirac field non-minimally coupled to the gravitational field within
the framework of the Poincar\'e gauge theory of gravity with torsion and
curvature. We study the subcase of "weak" gravity, that is, the gravitational
Lagrangian depends only linearly on the curvature and quadratically on the
torsion. We include all pieces in curvature and torsion that are of odd parity.
The second field equation of gravity is derived by varying the Lorentz
connection. We solve it with respect to the torsion and decompose the first
field equation of gravity and the Dirac equation into Einsteinian pieces and
post-Riemannian terms.
