  We introduce and study a construction of higher derived brackets generated by
a (not necessarily inner) derivation of a Lie superalgebra. Higher derived
brackets generated by an element of a Lie superalgebra were introduced in our
earlier work. Examples of higher derived brackets naturally appear in geometry
and mathematical physics. From a totally different viewpoint, we show that
higher derived brackets arise when one wants to turn the inclusion map of a
subalgebra of a differential Lie superalgebra, with a given complementary
subalgebra, into a fibration. (For a non-Abelian complementary subalgebra, this
leads to a generalization of $L_{\infty}$-algebras with dropped or weakened
(anti)symmetry of the brackets.)
