Commutativity-preserving maps on Lie ideals of prime algebras

One of the most studied linear preserver problems is the problem of characterizing maps preserving commutativity. In this paper we study maps preserving commutativity on a Lie ideal L of a prime algebra A. We show by an example that in general we cannot expect the same standard conclusion as in the case when L=A. Therefore we confine ourselves to some special classes of algebras where the usual result can be proved.