  This paper studies the self-dual Einstein-Dirac theory. A generalization is
obtained of the Jacobson-Smolin proof of the equivalence between the self-dual
and Palatini purely gravitational actions. Hence one proves equivalence of
self-dual Einstein-Dirac theory to the Einstein-Cartan-Sciama-Kibble-Dirac
theory. The Bianchi symmetry of the curvature, core of the proof, now contains
a non-vanishing torsion. Thus, in the self-dual framework, the extra terms
entering the equations of motion with respect to the standard Einstein-Dirac
field equations, are neatly associated with torsion.
