  In the last few years, there has been significant interest in understanding
the stationary comparison version of the first law of black hole mechanics in
the vielbein formulation of gravity. Several authors have pointed out that to
discuss the first law in the vielbein formulation one must extend the Iyer-Wald
Noether charge formalism appropriately. Jacobson and Mohd [arXiv:1507.01054]
and Prabhu [arXiv:1511.00388] formulated such a generalisation for symmetry
under combined spacetime diffeomorphisms and local Lorentz transformations. In
this paper, we apply and appropriately adapt their formalism to
four-dimensional gravity coupled to a Majorana field and to a Rarita-Schwinger
field. We explore the first law of black hole mechanics and the construction of
the Lorentz-diffeomorphism Noether charges in the presence of fermionic fields,
relevant for simple supergravity.
