  We get compact expressions for the Baker--Campbell--Hausdorff series $Z =
\log(\e^X \, \e^Y)$ in terms of right-nested commutators. The reduction in the
number of terms originates from two facts: (i) we use as a starting point an
explicit expression directly involving independent commutators and (ii) we
derive a complete set of identities arising among right-nested commutators. The
procedure allows us to obtain the series with fewer terms than when expressed
in the classical Hall basis at least up to terms of grade 10.
