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Let $G$ be a connected affine algebraic group, and $T$ be a maximal torus in $G$. We know that, $T$ is contained in some Borel subgroup $B$ of $G$. My question is:

Is it true that $C_{G}(T)=C_{B}(T)$?. Here, $C_{G}(T)$ denote the centraliser of $T$ in $G$.

Now, it is clear that $T\subseteq C_{G}(T)$. But, after that I have no clue how to proceed. Thanks in advance for any kind of help.

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Note that $C_G(T)\cap B$ is a Borel subgroup of $C_G(T).$ A proof of this is in Corollary of Section 22.4 in Humphreys' book. As $C_G(T)$ is nilpotent, $C_G(T)\cap B$ is also nilpotent. Thus, $C_G(T)\cap B = C_G(T)$. Thus, $C_G(T)$ is contained in $B$.

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