Quasi-derivations and QD-algebroids

Axioms of Lie algebroid are discussed. In particular, it is shown that a Lie QD-algebroid (i.e. a Lie algebra bracket on the C∞(M)-module ɛ of sections of a vector bundle E over a manifold M which satisfies [X, ƒ Y] = ƒ [X, Y] + A (X, ƒ)Y for all X, Y ε ɛ, ƒ ε C∞(M), and for certain A (X, ƒ) ε C∞(M)) is a Lie algebroid if rank (E) > 1, and is a local Lie algebra in the sense of Kirillov if E is a line bundle. Under a weak condition also the skew-symmetry of the bracket is relaxed.