Images of Lie Polynomials on simple Lie algebras

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image.