Let L be a Lie algebra over a commutative unital ring F containing 1 2. If L is perfect and centerless, then every skew-symmetric biderivation: L L L is of the form (x, y) = (x, y) for all x, y L, where Cent (L), the centroid of L. Under a milder assumption that c, [L, L] = 0 implies c = 0, every commuting linear map from L to L lies in Cent (L). These two results are special cases of our main theorems which concern biderivations and commuting linear maps having their ranges in an L-module. We provide a variety of examples, some of them showing the necessity of our assumptions and some of them showing that our results cover several results from the literature.
Bresar et al. (Mon,) studied this question.