[proofplan] We place $S$ inside a maximal torus $T$ of $G$ and analyze the centralizer using the root decomposition of the complexified [Lie algebra](/page/Lie%20Algebra) relative to $T$. The Lie algebra of $C_G(S)$ consists exactly of the toral directions together with those root directions on which every element of $S$ acts as the identity. A standard root-subgroup generation theorem then identifies the full centralizer, not only its identity component, with the connected subgroup generated by $T$ and those root subgroups. Since that subgroup is connected, the centralizer is connected. [/proofplan] [step:Choose a maximal torus containing $S$] Since $S$ is a torus in the compact connected Lie group $G$, it is compact, connected, abelian, and closed. By the maximal torus existence theorem for compact connected Lie groups, there exists a maximal torus $T\le G$ such that $S\le T$; equivalently, after extending the abelian compact connected subgroup $S$ to a maximal one, we obtain such a $T$. Let $\mathfrak g$ denote the Lie algebra of $G$, let $\mathfrak t$ denote the Lie algebra of $T$, and let $\mathfrak s$ denote the Lie algebra of $S$. Let \begin{align*} \mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C \end{align*} denote the complexified Lie algebra. [guided] The first structural move is to put the smaller torus $S$ inside a maximal torus. This is possible because $S$ is compact, connected, and abelian, and the [maximal torus theorem](/theorems/9713) for compact connected Lie groups says that compact connected abelian subgroups can be enlarged to maximal tori. We fix such a maximal torus $T\le G$ with $S\le T$. We write $\mathfrak g$ for the Lie algebra of $G$, $\mathfrak t$ for the Lie algebra of $T$, and $\mathfrak s$ for the Lie algebra of $S$. Since the root decomposition is naturally stated after complexification, define \begin{align*} \mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C. \end{align*} The point of choosing $T$ is that the action of $S$ on $\mathfrak g$ by the adjoint representation can be read off from the roots of $G$ relative to $T$. [/guided] [/step] [step:Identify the Lie algebra fixed by $S$] Write $X^*(T):=\operatorname{Hom}_{\mathrm{Lie}}(T,S^1)$ for the character group of $T$. For $t\in T$ and $s\in S$, write $\operatorname{Ad}_t:=\operatorname{Ad}(t)$ and $\operatorname{Ad}_s:=\operatorname{Ad}(s)$ for the adjoint action on $\mathfrak g_{\mathbb C}$. Let $\Phi\subset X^*(T)$ be the root system of $G$ with respect to $T$. By the root space decomposition for compact connected Lie groups, \begin{align*} \mathfrak g_{\mathbb C}=\mathfrak t_{\mathbb C}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha, \end{align*} where $\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C$ and where \begin{align*} \mathfrak g_\alpha:=\{X\in\mathfrak g_{\mathbb C}:\operatorname{Ad}_t(X)=\alpha(t)X\text{ for every }t\in T\} \end{align*} is the $\alpha$-root space. Let \begin{align*} \Phi_S:=\{\alpha\in\Phi:\alpha(s)=1\text{ for every }s\in S\} \end{align*} be the set of roots whose restriction to $S$ is trivial. Since $S\le T$, the adjoint action of every $s\in S$ is trivial on $\mathfrak t_{\mathbb C}$ and acts on $\mathfrak g_\alpha$ by the scalar $\alpha(s)$. Therefore the complexified Lie algebra of the centralizer is \begin{align*} \operatorname{Lie}(C_G(S))_{\mathbb C} = \mathfrak t_{\mathbb C}\oplus\bigoplus_{\alpha\in\Phi_S}\mathfrak g_\alpha. \end{align*} Indeed, an element $X\in\mathfrak g_{\mathbb C}$ lies in the complexified Lie algebra of $C_G(S)$ exactly when $\operatorname{Ad}_s(X)=X$ for every $s\in S$, and the displayed direct-sum decomposition makes this fixed-point condition componentwise. [guided] We now translate the group-theoretic centralizer condition into a Lie-algebraic fixed-point condition. The group $S$ acts on $\mathfrak g_{\mathbb C}$ by the complex-linear extension of the adjoint representation: \begin{align*} \operatorname{Ad}_s:\mathfrak g_{\mathbb C}\to\mathfrak g_{\mathbb C} \end{align*} for each $s\in S$. The root decomposition relative to the maximal torus $T$ gives \begin{align*} \mathfrak g_{\mathbb C}=\mathfrak t_{\mathbb C}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha, \end{align*} where \begin{align*} \mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C \end{align*} and \begin{align*} \mathfrak g_\alpha:=\{X\in\mathfrak g_{\mathbb C}:\operatorname{Ad}_t(X)=\alpha(t)X\text{ for every }t\in T\}. \end{align*} Because $S\le T$, every element of $S$ acts as the identity on $\mathfrak t_{\mathbb C}$. On the root space $\mathfrak g_\alpha$, the element $s\in S$ acts by multiplication by $\alpha(s)$. Thus a vector in $\mathfrak g_\alpha$ is fixed by every $s\in S$ exactly when $\alpha(s)=1$ for every $s\in S$. This motivates the definition \begin{align*} \Phi_S:=\{\alpha\in\Phi:\alpha(s)=1\text{ for every }s\in S\}. \end{align*} The Lie algebra of $C_G(S)$ is the fixed-point Lie algebra of this adjoint action: \begin{align*} \operatorname{Lie}(C_G(S))=\{X\in\mathfrak g:\operatorname{Ad}_s(X)=X\text{ for every }s\in S\}. \end{align*} After complexifying and using the directness of the root decomposition, the fixed-point condition is checked on each summand. Hence \begin{align*} \operatorname{Lie}(C_G(S))_{\mathbb C} = \mathfrak t_{\mathbb C}\oplus\bigoplus_{\alpha\in\Phi_S}\mathfrak g_\alpha. \end{align*} This step is where the torus hypothesis is essential: because $S$ lies in $T$, the adjoint action is simultaneously diagonalized by the root decomposition. [/guided] [/step] [step:Build the connected subgroup generated by the torus and the fixed root subgroups] For each root $\alpha\in\Phi$, let $G_\alpha\le G$ denote the connected root subgroup whose complexified Lie algebra is \begin{align*} \operatorname{Lie}(G_\alpha)_{\mathbb C} = \mathfrak g_\alpha\oplus\mathfrak g_{-\alpha}\oplus[\mathfrak g_\alpha,\mathfrak g_{-\alpha}]. \end{align*} This is the standard compact rank-one subgroup attached to $\alpha$. Define $H\le G$ to be the subgroup generated by $T$ and all $G_\alpha$ with $\alpha\in\Phi_S$: \begin{align*} H:=\langle T,\ G_\alpha\text{ for }\alpha\in\Phi_S\rangle. \end{align*} The group $T$ is connected, and each $G_\alpha$ is connected. Since all these connected subgroups contain the identity element, the subgroup generated by them is connected. Hence $H$ is connected. For every $\alpha\in\Phi_S$, the restriction of $\alpha$ to $S$ is trivial, and the corresponding rank-one root subgroup is centralized by $S$. Also $T$ centralizes $S$ because $T$ is abelian and $S\le T$. Therefore \begin{align*} H\le C_G(S). \end{align*} By construction and by the root-subgroup Lie algebra formula, \begin{align*} \operatorname{Lie}(H)_{\mathbb C} = \mathfrak t_{\mathbb C}\oplus\bigoplus_{\alpha\in\Phi_S}\mathfrak g_\alpha. \end{align*} Comparing with the previous step gives \begin{align*} \operatorname{Lie}(H)=\operatorname{Lie}(C_G(S)). \end{align*} [/step] [step:Use root subgroup generation to exclude extra components] We now use the standard centralizer-generation theorem for compact connected Lie groups: if $S\le T$ is a subtorus of a maximal torus, then \begin{align*} C_G(S)=\langle T,\ G_\alpha\text{ for }\alpha\in\Phi\text{ with }\alpha|_S=1\rangle. \end{align*} Equivalently, the centralizer of a subtorus is generated by the maximal torus and the root subgroups on which the subtorus acts as the identity. This is the root subgroup generation theorem for centralizers of subtori. Applying this theorem to the subtorus $S\le T$ gives \begin{align*} C_G(S)=H. \end{align*} Since $H$ is connected, it follows that $C_G(S)$ is connected. [guided] At this point we have proved that $H$ is a connected subgroup of $C_G(S)$ and that $H$ has the same Lie algebra as $C_G(S)$. That proves that $H$ is the identity component of $C_G(S)$, but it does not by itself rule out extra connected components. This is the delicate point in the theorem. The required structural input is the root subgroup generation theorem for centralizers of subtori in compact connected Lie groups. It states that, for a maximal torus $T\le G$ and a subtorus $S\le T$, \begin{align*} C_G(S)=\langle T,\ G_\alpha\text{ for }\alpha\in\Phi\text{ with }\alpha|_S=1\rangle. \end{align*} The hypotheses match our situation: $G$ is compact and connected, $T$ is a maximal torus of $G$, and $S\le T$ is a subtorus. The right-hand side is exactly the subgroup $H$ we defined: \begin{align*} H=\langle T,\ G_\alpha\text{ for }\alpha\in\Phi_S\rangle, \end{align*} where \begin{align*} \Phi_S=\{\alpha\in\Phi:\alpha(s)=1\text{ for every }s\in S\}. \end{align*} Therefore \begin{align*} C_G(S)=H. \end{align*} Since $H$ was already shown to be connected, the equality proves that $C_G(S)$ has no additional components. Hence $C_G(S)$ is connected. [/guided] [/step]