Theorem: Assume k does not contain a non-zero ideal of g. Then the following statements are equivalent:

1. p:z(g)\otimes z(k)-->U(g)^k is an isomorphism;
2. U(g)^k is commutative;
3. the pair (g,k) is either (sl(n), gl(n-1)) n>=2 or (so(n), so(n-1)) n>=4.

• Allan Cooper: The classifying ring of groups whose classifying ring is commutative. MIT-Thesis (1975) unpublished.
  Here, (3. implies 1.) is proved for the first time. Cooper attributes the proof of (2. implies 3.) to Bertram Kostant but nothing is written up.
• Roger Howe: Some highly symmetrical dynamical systems. Preprint (undated)
  Here, (3. implies 1.) is proved basically in the same way as in my paper.
• Abdel Ilah Benabdallah: Generateurs de l'algebre U(G)^K avec G=SO(m) ou SO_0(1,m-1) et K=SO(m-1). Bull. Soc. Math. France 111 (1983) 303-326.
  This paper contains a proof of (3. implies 1.) for (so(n),so(n-1)).
• Isabelle Sezionale Basilicato: Structure de U(g)^k pour certains groupes classiques. C. R. Acad. Sci. Paris Ser. I Math. 297 (1983) 13-16.
  This paper contains a proof of (3. implies 1.) for (sl(n),gl(n-1)).
• Kenneth Johnson: The centralizer of a Lie algebra in an enveloping algebra. J. reine angew. Math. 395 (1989) 196-201.
  Yet another proof of (3. implies 1.) for (sl(n),gl(n-1)). This was the only paper on this problem I was aware of. In fact, it triggered my interest.
• Nicolas Andruskiewitsch: On the complicatedness of the pair (g,K). Rev. Math. Univ. Compl. Madrid 2 (1989) 13-28.
  This paper contains to my knowledge the first published proof of (2. implies 3.) (allthough implicitely).