# Let $C=C_G(H)$, $\mathfrak{c} \subseteq \mathfrak{c}_\mathfrak{g}(\mathfrak{h})$

Let $H$ be a closed subgroup of the algebraic group $G$, $C=C_G(H)$. Prove that $\mathfrak{c} \subseteq \mathfrak{c}_\mathfrak{g}(\mathfrak{h})=\{ \mathbb{x} \in \mathfrak{g} : [\mathbb{x}, \mathfrak{h}] =0\}$.

Here, $\mathfrak{g}$,$\mathfrak{h}$ and $\mathfrak{c}$ are the Lieg algebras of $G$, $H$ and $C$ respectively.

If $I$ is the ideal of $K[G]$ vanishing on $H$, then $\mathfrak{h}$ is the subalgebra of $\mathfrak{g}$ consists of elements mapping $I$ to $0$. But what is the connection between the ideals vanishing on $H$ and $C$ making the relation $\mathfrak{c} \subseteq \mathfrak{c}_\mathfrak{g}(\mathfrak{h})$ hold? Or is there another way to prove this proposition? Many thanks.

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If you go to Milne's notes on Algebraic Groups, see Prop 2.23 on pg 261. See jmilne.org/math/CourseNotes/ala.html – John M Jul 17 '11 at 19:06
This is very useful. Thank you very much. – ShinyaSakai Jul 22 '11 at 13:14