Constructing an even Poisson algebra from a Gerstenhaber algebra by means of an odd derivation of square 0 is shown to be possible in the category of Loday algebras (algebras with a non-skew-symmetric bracket, generalizing the Lie algebras, heretofore called Leibniz algebras in the literature). Such “derived brackets” give rise to Lie brackets on certain quotient spaces, and also on certain Abelian subalgebras. The construction of these derived brackets explains the origin of the Lie bracket on the space of co-exact differential forms on a Poisson manifold. We further examine the derived brackets on the space of cochains of an associative or Lie algebra. Finally, we relate the previous result to various generalizations of the notion of BV-algebra.