Exercise 12, § 9

Let $U^0$ be the commutant of $\mathfrak{h}$ in $U(\mathfrak{g})$. If $\mathfrak{g}$ is of rank $1, U^0$ is commutative. If $\mathfrak{g}$ is of rank $\geq 2, U^0$ admits simple representations of arbitrarily large finite dimension. (Use Exerc. 11 and Exerc. $23a$) of §7.)