On Dixmier's fourth problem

Let L be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. Denote by U(L) its enveloping algebra with quotient division ring D(L). In 1974, at the end of his book "Algebres enveloppantes", Jacques Dixmier listed 40 open problems, of which the fourth one asked if the center Z(D(L)) is always a purely transcendental extension of k. We show this is the case if L is algebraic whose Poisson semi-center Sy(L) is a polynomial algebra over k. This can be applied to many (bi)parabolic subalgebras of semi-simple Lie algebras. We also provide a survey of Lie algebras for which Dixmier's problem is known to have a positive answer. This includes all Lie algebras of dimension at most 8. We prove this is also true for all 9-dimensional algebraic Lie algebras. Finally, we improve Theorem 53 of [45].