  In this note we provide a new construction of BCJ dual-trace factor using the
kinematic algebra proposed in arXiv:1105.2565 and arXiv:1212.6168. Different
from the construction given in arXiv:1304.2978 based on the proposal of
arXiv:1103.0312, the method used in this note exploits the adjoint
representation of kinematic algebra and the use of inner product in dual space.
The dual-trace factor defined in this way naturally satisfies cyclic symmetry
condition but not KK-relation, just like the trace of U(N) Lie algebra
satisfies cyclic symmetry condition, but not KK-relation. In other words the
new construction naturally leads to formulation sharing more similarities with
the color decomposition of Yang-Mills amplitude.
