Deriving spin-1 quartic interaction vertices from closure of the Poincar\'e algebra

We derive the quartic interaction vertex of pure Yang-Mills theory by demanding closure of the light-cone Poincar'e algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We then prove that corrections to the spin generator, for spin one at this order, vanish. We comment briefly on higher spin fields in this context.