  Building on our proposal in arXiv:2405.06629, we present in detail the
construction of the extended phase space for Yang-Mills at null infinity,
containing the asymptotic symmetries and the charges responsible for
sub$^n$-leading soft theorems at all orders. The generality of the procedure
allows it to be directly applied to the computation of both tree and loop-level
soft limits. We also give a detailed study of Yang-Mills equations under the
radial expansion, giving a thorough construction of the radiative phase space
for decays compatible with tree-level amplitudes for both light-cone and radial
gauges. This gives rise to useful recursion relations at all orders between the
field strength and the vector gauge coefficients. We construct the
sub$^n$-leading charges recursively, and show a hierarchical truncation such
that each charge subalgebra is closed, and their action in the extended phase
space is canonical. We relate these results with the infinite-dimensional
algebras that have been recently introduced in the context of conformal field
theories at null infinity.
