Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces

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 section s of Courant algebroids and on the "omni-Lie algebras" recently introduced by the second author.