  The Leibniz bracket of an operator on a (graded) algebra is defined and some
of its properties are studied. A basic theorem relating the Leibniz bracket of
the commutator of two operators to the Leibniz bracket of them, is obtained.
Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie
algebra structure. In particular, those algebras generated by the Leibniz
bracket of the divergence and the Laplacian operators on the exterior algebra
are considered, and the expression of the Laplacian for the product of two
functions is generalized for arbitrary exterior forms.
