  Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra,
for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to
g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove
that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any
Chevalley algebra $g$. This result can be viewed as a weak infinitesimal
counterpart of Borel's theorem on the dominancy of the word map on connected
semisimple algebraic groups.
  We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more
generally, for their linear combinations, this map is, moreover, surjective
onto the set of noncentral elements of $g$ provided that the ground field $K$
is big enough, and show that for monomials of large degree the image of this
map contains no nonzero central elements.
  We also discuss consequences of these results for polynomial maps of
associative matrix algebras.
