  We show that the skew-symmetrized product on every Leibniz algebra E can be
realized on a reductive complement to a subalgebra in a Lie algebra. As a
consequence, we construct a nonassociative multiplication on E which, when E is
a Lie algebra, is derived from the integrated adjoint representation. We apply
this construction to realize the bracket operations on the sections of Courant
algebroids and on the ``omni-Lie algebras'' recently introduced by the second
author.
