Tree-level n-point gauge-theory amplitudes with n − 2k gluons and k pairs of (massless or massive) particles in the fundamental (or other) representation of the gauge group are invariant under a set of symmetries that act as momentum-dependent shifts on the color factors in the cubic decomposition of the amplitude. These symmetries lead to gauge-invariant constraints on the kinematic numerators. They also directly imply the BCJ relations among the Melia-basis primitive amplitudes previously obtained by Johansson and Ochirov.
Brown, Robert W. and Naculich, Stephen G., "Color-factor symmetry and BCJ relations for QCD amplitudes" (2016). Physics Faculty Publications. 106.