A primitive ideal determines a central character

Given: A finite-dimensional complex Lie algebra $\mathfrak{g}$, a simple left $U(\mathfrak{g})$-module $M$, and the primitive ideal $I={\mathrm{Ann}}_{U(\mathfrak{g})}(M)$.

$U(\mathfrak{g})$ is countable-dimensional for finite-dimensional $\mathfrak{g}$, so Dixmier's lemma applies: ${\mathrm{End}}_{U(\mathfrak{g})}(M)=\mathbb{C}$, every central element acts on $M$ by a scalar, and these scalars form a unital $\mathbb{C}$-algebra homomorphism ${\chi }_I\text{\hspace{0.17em}⁣}:Z(U(\mathfrak{g}))\to \mathbb{C}$ with $zm={\chi }_I(z)m$ (Dixmier's lemma: endomorphisms of a simple module over a countable-dimensional algebra, Central character of a Lie algebra module).

A unital homomorphism $\chi \text{\hspace{0.17em}⁣}:Z(U(\mathfrak{g}))\to \mathbb{C}$ has image $\mathbb{C}$, so $Z(U(\mathfrak{g}))\mathrm{/}\ker \chi \cong \mathbb{C}$ is a field and $\ker \chi$ is a maximal ideal; maximal means maximal among proper ideals ($R\mathrm{/}M$ is a field if and only if $M$ is a maximal ideal, Prime ideals and maximal ideals in a commutative ring).

$I$ is a two-sided ideal equal to the kernel of the action; $M\mathrm{\ne }0$ and $1$ acts as the identity, so $I$ is proper (The annihilator of a module over an enveloping algebra, Primitive ideals of an enveloping algebra, Simple module: a nonzero module with no proper nonzero submodule).