  We present a systematic method to determine BCJ numerators for one-loop
amplitudes that explores the global constraints on the loop momentum
dependence. We apply this method to amplitudes in N=4 gauge theory, working out
detailed examples up to seven points in both the MHV and the NMHV sectors. The
structure of Jacobi identities between BCJ numerators is seen to be closely
connected to that of algebraic integrand reductions. We discuss the
consequences for one-loop N=8 supergravity amplitudes obtained through the
double copy prescription. Moreover, in the MHV sector, we show how to obtain
simple BCJ box numerators using a conjectured relationship to amplitudes in
self-dual gauge theory. We also introduce simpler trace-type formulas for
integrand reductions.
