Abstract
In this paper, we find conditions under which the bracket defined by a graded derivation on a Lie superalgebra (g, [, ]) is skew-supersymmetry and satisfies the super Jacobi identity, so it defines the structure of a Lie superalgebra on g.
In the case of the algebra of differential forms on a supermanifold, we study the graded commutator of graded derivations, graded skew-derivations and a graded derivation, with another graded skew-derivation of the superalgebra of differential forms on a supermanifold.