  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.
