  The duality between color and kinematics present in scattering amplitudes of
Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie
algebra that controls the gauge theory. While associated BCJ numerators are
known on closed forms to any multiplicity at tree level, the kinematic algebra
has only been partially explored for the simplest of four-dimensional
amplitudes: up to the MHV sector. In this paper we introduce a framework that
allows us to characterize the algebra beyond the MHV sector. This allows us to
both constrain some of the ambiguities of the kinematic algebra, and better
control the generalized gauge freedom that is associated with the BCJ
numerators. Specifically, in this paper, we work in dimension-agnostic notation
and determine the kinematic algebra valid up to certain ${\cal
O}\big((\varepsilon_i \cdot \varepsilon_j)^2\big)$ terms that in four
dimensions compute the next-to-MHV sector involving two scalars. The kinematic
algebra in this sector is simple, given that we introduce tensor currents that
generalize standard Yang-Mills vector currents. These tensor currents controls
the generalized gauge freedom, allowing us to generate multiple different
versions of BCJ numerators from the same kinematic algebra. The framework
should generalize to other sectors in Yang-Mills theory.
