  In this article we present the algebraic rearrangement, or matrix inversion
of the Dirac equation in a curved Riemann-Cartan spacetime with torsion, the
presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the
Dirac field. We then demonstrate how the inversion leads to a reformulation of
the fully non-linear and self-interactive Einstein-Cartan-Dirac field equations
in terms of Dirac bilinears. It has been known for some decades that the Dirac
equation for charged fermions interacting with an electromagnetic field can be
algebraically inverted, so as to obtain an explicit rational expression of the
four-vector potential of the gauge field in terms of the spinors. Substitution
of this expression into Maxwell's equations yields the bilinear form of the
self-interactive Maxwell-Dirac equations. In the present (purely gravitational)
case, the inversion process yields \emph{two} rational four-vector expressions
in terms of Dirac bilinears, which act as gravitational analogues of the
electromagnetic vector potential. These "potentials" also appear as irreducible
summand components of the connection, along with a traceless residual term of
mixed symmetry. When taking the torsion field equation into account, the
residual term can be written as a function of the object of anholonomity. Using
the local tetrad frame associated with observers co-moving with the Dirac
matter, a generic vierbein frame can described in terms of four Dirac bilinear
vector fields, normalized by a scalar and pseudoscalar field. A corollary of
this is that in regions where the Dirac field is non-vanishing, the
self-coupled Einstein-Cartan-Dirac equations can in principle be expressed in
terms of Dirac bilinears only.
