  We use covariant phase space methods to study the metric and tetrad
formulations of General Relativity in a manifold with boundary and compare the
results obtained in both approaches. Proving their equivalence has been a
long-lasting problem that we solve here by using the cohomological approach
provided by the relative bicomplex framework. This setting provides a clean and
ambiguity-free way to describe the solution spaces and associated symplectic
structures. We also compute several relevant charges in both schemes and show
that they are equivalent, as expected.
