  We write down closed formulas for all necessary steps to obtain multiparticle
super Yang--Mills superfields in the so-called BCJ gauge. The superfields in
this gauge have obvious applications in the quest for finding BCJ-satisfying
representations of amplitudes. As a benefit of having these closed formulas, we
identify the explicit finite gauge transformation responsible for attaining the
BCJ gauge. To do this, several combinatorial maps on words are introduced and
associated identities rigorously proven.
