Solvable symmetric Poisson algebras and their derived lengths

Let L be a Lie algebra over a eld of positive characteristic and let S(L) and s(L) denote, respectively, the symmetric Poisson algebra and the truncated symmetric Poisson algebra of L. As a natural continuation of the work by Monteiro Alves and Petrogradsky, we investigate the structure of L when S(L) or s(L) is solvable. We first disprove a conjecture stated in [9] about solvability of S(L) (and s(L)) in characteristic 2. Next, the derived lengths of s(L) are studied. In particular, we provide bounds for the derived lengths of s(L), establish when s(L) is metabelian, and characterize truncated symmetric Poisson algebras of minimal derived length