On Herstein's Lie map conjectures, I

We describe subjective Lie homomorphisms from Lie ideals of skew elements of algebras with involution onto noncentral Lie ideals (factored by their centers) of skew elements of prime algebras D with involution, provided that char( D) not equal 2 and D is not PI of low degree. This solves the last remaining open problem of Herstein on Lie isomorphisms module cases of PI rings of low degree. A more general problem on maps preserving any polynomial is also discussed.