  Dual gravitational charges have been recently computed from the Holst term in
tetrad variables using covariant phase space methods. We highlight that they
originate from an exact 3-form in the tetrad symplectic potential that has no
analogue in metric variables. Hence there exists a choice of the tetrad
symplectic potential that sets the dual charges to zero. This observation
relies on the ambiguity of the covariant phase space methods. To shed more
light on the dual contributions, we use the Kosmann variation to compute
(quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a
formula that illustrates comprehensively why the dual contribution to the
Hamiltonian charges: (i) vanishes for exact isometries and asymptotic
symmetries at spatial infinity; (ii) persists for asymptotic symmetries at
future null infinity, in addition to the usual BMS contribution. Finally, we
point out that dual gravitational charges can be equally derived using the
Barnich-Brandt prescription based on cohomological methods, and that the same
considerations on asymptotic symmetries apply.
