  Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie
algebraic structure that suggest the existence of a hidden infinite-dimensional
kinematic algebra. Explicitly realizing such a kinematic algebra is a
longstanding open problem that only has had partial success for simple helicity
sectors. In past work, we introduced a framework using tensor currents and
fusion rules to generate BCJ numerators of a special subsector of NMHV
amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly
realize a kinematic algebra for all NMHV amplitudes. Master numerators are
obtained directly from the algebraic rules and through commutators and
kinematic Jacobi identities other numerators can be generated. Inspecting the
output of the algebra, we conjecture a closed-form expression for the master
BCJ numerator up to any multiplicity. We also introduce a new method, based on
group algebra of the permutation group, to solve for the generalized gauge
freedom of BCJ numerators. It uses the recently introduced binary BCJ relations
to provide a complete set of NMHV kinematic numerators that consist of pure
gauge.
