  In [15] we proposed a generalization of the BMS group G which is a semidirect
product of supertranslations and smooth diffeomorphisms of the conformal
sphere. Although an extension of BMS, G is a symmetry group of asymptotically
flat space times. By taking G as a candidate symmetry group of the quantum
gravity S-matrix, we argued that the Ward identities associated to the
generators of Diff(S^2) were equivalent to the Cachazo-Strominger subleading
soft graviton theorem. Our argument however was based on a proposed definition
of the Diff(S^2) charges which we could not derive from first principles as G
does not have a well defined action on the radiative phase space of gravity.
Here we fill this gap and provide a first principles derivation of the
Diff(S^2) charges. The result of this paper, in conjunction with the results of
[4, 15] prove that the leading and subleading soft theorems are equivalent to
the Ward identities associated to G.
