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

We show that the skew-symmetrized product on every Leibniz algebra ε can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on ε which, when ε 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.