  One important discovery in recent years is that the total amplitude of gauge
theory can be written as BCJ form where kinematic numerators satisfy Jacobi
identity. Although the existence of such kinematic numerators is no doubt, the
simple and explicit construction is still an important problem. As a small
step, in this note we provide an algebraic approach to construct these
kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi
identity is manifestly satisfied. The corresponding color ordered amplitudes
satisfy off-shell KK-relation and off-shell BCJ relation similar to the color
ordered scalar theory. Using our construction, the dual DDM form is also
established.
