  Let L be a finite dimensional Lie algebra over an algebraically closed field
k of characteristic zero. We provide necessary and also some sufficient
conditions in order for its Poisson center and semi-center to be polynomial
algebras over k. This occurs for instance if L is quadratic of index 2 with L
not equal to [L,L] and also if L is nilpotent of index at most 2. The converse
holds for filiform Lie algebras of type Ln, Qn, Rn and Wn. We also show how
Dixmier's fourth problem for an algebraic Lie algebra L can be reduced to that
of its canonical truncation. Moreover, Dixmier's statement holds for all Lie
algebras of dimension at most eight. The nonsolvable, indecomposable ones among
them possess a polynomial Poisson center and semi-center.
