Let \(\mathfrak {g}\) be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. Denote by \(U(\mathfrak {g})\) its enveloping algebra with quotient division ring \(D(\mathfrak {g})\). In 1974, at the end of his book “Algèbres enveloppantes”, Jacques Dixmier listed 40 open problems, of which the fourth one asked if the center \(Z(D(\mathfrak {g}))\) is always a purely transcendental extension of k. We show this is the case if \(\mathfrak {g}\) is algebraic whose Poisson semi-center \(Sy(\mathfrak {g})\) is a polynomial algebra over k. This can be applied to many biparabolic (seaweed) 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 the statement of Theorem 53 of Ooms (J. Algebra 477, 95–146, 2017).

令\（\ mathfrak {g} \）是特征为零的代数闭合域k上的有限维李代数。用\（U（\ mathfrak {g}）\）表示其包络代数与商除环\（D（\ mathfrak {g}）\）。1974年，雅克·迪克西米尔（Jacques Dixmier）在他的著作《阿尔及尔的信封》（Algèbresenveloppantes）的结尾处列出了40个未解决的问题，其中第四个问题询问中心\（Z（D（\（mathfrak {g}））\）是否始终是纯先验k的扩展。我们证明如果\（\ mathfrak {g} \）是其Poisson半中心\（Sy（\ mathfrak {g}）\）是k上的多项式代数的代数。这可以应用于半简单李代数的许多双抛物（海藻）子代数。我们还提供了一个Lie代数的调查，已知Dixmier问题对此给出了肯定的答案。这包括最多8个维的所有Lie代数。我们证明这对于所有9维代数Lie代数也是正确的。最后，我们提高奥姆斯的定理53（J.代数的声明477，95-146，2017）。