Commutativity of the Centralizer of a Subalgebra in a Universal Enveloping Algebra

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic zero, and let $\mathfrak{h}$ be an algebraic subalgebra of the tangent Lie algebra $\mathfrak{g}$ of $G$. We find all subalgebras $\mathfrak h$ that have no nontrivial characters and whose centralizers $\mathfrak{U}(\mathfrak{g})^\mathfrak{h}$ and $P(\mathfrak{g})^{\mathfrak{h}}$ in the universal enveloping algebra $\mathfrak{U}\mathfrak{g})$ and in the associated graded algebra $P(\mathfrak{g})$, respectively, are commutative. For all these subalgebras, we prove that ${\mathfrak U}\mathfrak{(g)}^{\mathfrak h}=\mathfrak{U(h)^h}\otimes\mathfrak{U(g)^g}$ and $P\mathfrak{(g)}^{\mathfrak h}=P\mathfrak{(h)^h}\otimes P\mathfrak{(g)^g}$. Furthermore, we obtain a criterion for the commutativity of $\mathfrak{U(g)^h}$ in terms of representation theory.