  Basing ourselves on the categorical notions of central extensions and
commutators in the framework of semi-abelian categories relative to a Birkhoff
subcategory, we study central extensions of Leibniz algebras with respect to
the Birkhoff subcategory of Lie algebras, called Lie-central extensions. We
obtain a six-term exact homology sequence associated to a Lie-central
extension. This sequence, together with the relative commutators, allows us to
characterize several classes of Lie-central extensions, such as Lie-trivial,
Lie-stem and Lie-stem cover, to introduce and characterize Lie-unicentral,
Lie-capable, Lie-solvable and Lie-nilpotent Leibniz algebras.
