  Similar to gravity, an infinite tower of symmetries generated by higher-spin
charges has been identified in Yang-Mills theory by studying collinear limits
or celestial operator products of gluons. This work aims to recover this loop
symmetry in terms of charge aspects constructed on the gluonic Fock space. We
propose an explicit construction for these higher spin charge aspects as
operators which are polynomial in the gluonic annihilation and creation
operators. The core of the paper consists of a proof that the charges we
propose form a closed loop algebra to quadratic order. This closure involves
using the commutator of the cubic order expansion of the charges with the
linear (soft) charge. Quite remarkably, this shows that this
infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills
theory. We provide a similar all spin proof in gravity for the so-called global
quadratic (hard) charges which form the loop wedge subalgebra of
$w_{1+\infty}$.
