Fermions are coupled to the Einstein-Cartan system in the canonical formulation, including the cosmological, the Barbero-Immirzi, and the nonminimal coupling constants. The resulting ten first-class constraints generate gauge transformations that are on-shell equivalent to spacetime diffeomorphisms and $\mathrm{SL}(2,\mathbb{C})$ transformations. The gravitational second-class constraints receive fermionic contributions, which can be implemented by use of Dirac brackets or by solving them directly. Furthermore, we identify new fermionic (second-class) constraints that are necessary to recover the Dirac-fermion theory by relating the momenta to the configuration variables on dynamical solutions; this fermionic phase-space reduction is accomplished by use of corresponding Dirac brackets. The theory remains well defined off the Dirac-Fermion constraint surface with four additional degrees of freedom. Discrete ($CPT$) symmetries as well as implications for canonical quantization and modified theories of gravity with fermions are discussed.

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• General relativity formalism