  We introduce a non-abelian exterior product of two crossed modules of Leibniz
algebra and investigate its relation to the low dimensional Leibniz homology.
Later this non-abelian exterior product is applied to the construction of eight
term exact sequence in Leibniz homology. Also its relationship to the universal
quadratic functor is established, which is applied to the comparison of the
second Lie and Leibniz homologies of a Lie algebra.
