A class of zero product determined Lie algebras

Let L be a Lie algebra over a field F. We say that L is zero product determined if, for every F-linear space V and every bilinear map φ:L×L→V, the following condition holds. If φ(x,y)=0 whenever [x,y]=0, then there exists a linear map f from [L,L] to V such that φ(x,y)=f([x,y]) for all x,y∈L. This article shows that every parabolic subalgebra p of a (finite-dimensional) simple Lie algebra defined over an algebraically closed field is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010) [9] to determine zero product derivations of p, and we obtain a definitive solution for the problem of describing two-sided commutativity-preserving maps on p.