  The Poisson brackets of the gravitational field at null infinity play a
pivotal role in establishing the equivalence between the Ward identities
involving BMS charges and the soft graviton theorem. In recent literature it
was noticed that, in order to reproduce the action of BMS transformations via
such Poisson brackets, one needs to add "ad-hoc" boundary terms in the
symplectic form. In this note we show that, introducing a suitable splitting of
the gravitational field in bulk and boundary degrees of freedom and using
techniques of covariant phase space formalism, it is possible to obtain the
correct Poisson brackets between the boundary fields without any additional
assumption. The same Poisson brackets are used to show that BMS charges
canonically generate BMS transformations on the gravitational phase space.
