  Symmetries of Einstein-Yang-Mills (EYM) amplitudes, together with the
recursive expansions, induce nontrivial identities for pure Yang-Mills
amplitudes. In the previous work \cite{Hou:2018bwm}, we have already proven
that the identities induced from tree level single-trace EYM amplitudes can be
precisely expanded in terms of BCJ relations. In this paper, we extend the
discussions to those identities induced from all tree level \emph{multi-trace}
EYM amplitudes. Particularly, we establish a refined graphic rule for
multi-trace EYM amplitudes and then show that the induced identities can be
fully decomposed in terms of BCJ relations.
