  We clarify the structure obtained in H\'elein and Vey's proposition for a
variational principle for the Einstein-Cartan gravitation formulated on a frame
bundle starting from a structure-less differentiable 10-manifold. The obtained
structure is locally equivalent to a frame bundle which we term "generalised
frame bundle". In the same time, we enrich the model with a Dirac spinor
coupled to the Einstein-Cartan spacetime. The obtained variational equations
generalise the usual Einstein-Cartan-Dirac field equations in that they are
shown to imply the usualy field equations when the generalised frame bundle is
a standard frame bundle.
