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Let $G$ be a connected complex semisimple Lie group, and let $\mathfrak{h}\subset\mathfrak{g}$ be a Cartan subalgebra (CSA). Let $H=\text{exp}(\mathfrak{h})\cong \mathfrak{h}/\Gamma\cong \mathbb{C}^\times\times\dots\times\mathbb{C}^\times$ where $\Gamma\subset\mathfrak{h}$ is the dual lattice of some lattice in $\mathfrak{h}^*$ sitting between the root lattice and the weight lattice.

My main question is: why is $C_G(H)=\{g\in G|gx=xg \text{ for all }x\in H\}$, the centralizer of $H$ in $G$, equal to $H$? Because $\mathfrak{h}$ is self-normalizing I know it has connected component of the identity equal to $H$, so my question is why is $C_G(H)$ connected?

Please don't cite the fact that $N(H)/H$ is isomorphic to the Weyl group unless you know a proof of that which doesn't use $C_G(H)=H$.

Any/all help/advice is greatly appreciated!

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