Lemma 2

Let S be a closed subgroup of T and Z(S) its normalizer in G.

(i) $R(Z(S)_0,T)$ is the set of $\alpha \in R(G,T)$ such that $\alpha (S) =\{1\}$.

(ii) The centre of $Z(S)_0$ is the intersection of the Ker$\alpha$ for $\alpha \in R(Z(S)_0,T)$.

(iii) If S is connected, Z(S) is connected.

The Lie algebra $L(Z(S))_{(\mathbf{C})}$ consists of the invariants of S on $\mathfrak{g}_{\mathbf{C}}$ (Chap. III, §9, no. 3, Prop. 8), and hence is the direct sum of $\mathfrak{t}_{\mathbf{C}}$ and the $\mathfrak{g}^{\alpha}$ for which $\alpha (S) =\{1\}$, hence (i). Assertion (ii) follows from Prop. 8 (no. 4), and assertion (iii) has already been proved (§2, no. 2, Cor. 5 of Th. 2).