  We show that the non-linear BMS$_5$ symmetry algebra of asymptotically flat
Einstein gravity in five dimensions, as well as the super-BMS$_4$ superalgebra
of asymptotically flat supergravity, can be redefined so as to take a direct
sum structure. In the new presentation of the (super-)algebra, angle-dependent
translations and angle-dependent supersymmetry transformations commute with the
(super-)Poincar\'e generators. We also explain in detail the structure and
charge-integrability of asymptotic symmetries with symmetry parameters
depending on the fields (through the charges themselves), a topic relevant for
nonlinear asymptotic symmetry algebras.
