  The asymptotic symmetry algebra of four-dimensional Einstein gravity in the
asymptotically flat context has been shown recently to be the direct sum of the
Poincar\'e algebra and of an infinite-dimensional abelian algebra (with central
charge) that includes the Bondi-Metzner-Sachs supertranslations. This result,
obtained within the Hamiltonian formalism, yields a supertranslation invariant
definition of the Lorentz generators (angular momentum and boosts). Definitions
of Lorentz generators free from the ``supertranslation ambiguities'' have also
been proposed recently at null infinity. We prove the equivalence of the two
approaches for redefining the charges.
