Proposition 8

Let $G$ be a finite-dimensional real or complex Lie group, $g$ its Lie algebra and $A$ a subset of $G$. Then $Z_G(A)$ is a Lie subgroup of $G$ with Lie algebra $\mathfrak{z}_g(A)$.

Suppose that $A$ consists of a single point $a$. Then $Z_G(A)$ is the set of fixed points of $\mathrm{Int}\, a$; hence $Z_G(A)$ is a Lie subgroup of $G$ and $\mathbf{L}(Z_G(A))$ is the set of fixed points of $\mathrm{Ad}\, a$, that is $\mathfrak{z}_g(A)$ (§ 3, no. 8, Corollary 1 to Proposition 29). The general case follows by means of § 6, no. 2, Corollary 3 to Proposition 1.