Equations in simple Lie algebras

We prove that for a given element P(X1,…,Xd) of the finitely generated free Lie algebra Ld, the induced map P:gd→g is dominant for any Chevalley algebra g, provided that K is of characteristic ≠2, and P is not an identity in sl(2,K). We prove that for the Engel monomials [[[X,Y],Y],…,Y] and for their linear combinations this map is surjective onto the set of non-central elements of g provided that the ground field K is big enough.