Characterizing additive ξ-Lie derivations of prime algebras by ξ-Lie zero products

Let A be a unital prime algebra with a nontrivial idempotent over a field F. For any scalar ξ∈F, all additive maps L:A→A satisfying [L(A),B]ξ+[A,L(B)]ξ=0 whenever [A,B]ξ=0 are characterized and the relation of L to the derivations are revealed, where [A,B]ξ=AB-ξBA is the ξ-Lie product of A,B∈A.