Let $${\mathcal{L}}$$ be a $${\mathcal{J}}$$ -subspace lattice on a Banach space X over the real or complex field $${\mathbb{F}}$$ with dim X ≥ 2 and Alg $${\mathcal{L}}$$ be the associated $${\mathcal{J}}$$ -subspace lattice algebra. For any scalar $${\xi \in \mathbb{F}}$$ , there is a characterization of any linear map L : Alg $${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$$ satisfying $${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$$ for any $${A, B \in{\rm Alg} {\mathcal{L}}}$$ with AB = 0 (rep. $${[A,B]_ \xi = AB - \xi BA = 0}$$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg $${\mathcal{L}}$$ is obtained.