Deriving spin-1 quartic interaction vertices from closure of the Poincaré algebra

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincaré algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We prove that there is no correction to the spin generator, for spin one, at this order. We comment briefly on higher spin fields in this context.