  We propose a kinematic algebra for the Bern-Carrasco-Johansson (BCJ)
numerators of tree-level amplitudes and form factors in Yang-Mills theory
coupled with bi-adjoint scalars. The algebraic generators of the algebra
contain two parts: the first part is simply the flavour factor of the
bi-adjoint scalars, and the second part that maps to non-trivial kinematic
structures of the BCJ numerators obeys extended quasi-shuffle fusion products.
The underlying kinematic algebra allows us to present closed forms for the BCJ
numerators with any number of gluons and two or more scalars for both on-shell
amplitudes and form factors that involve an off-shell operator. The BCJ
numerators constructed in this way are manifestly gauge invariant and obey many
novel relations that are inherited from the kinematic algebra.
