Corollary 2

Let H be a connected closed subgroup of G containing T, and let $W^H_G(T)$ be the stabilizer in $W_G(T)$ of the subset $R(H,T)$ of R. The group $N_G(H)/H$ is isomorphic to the quotient group $W^H_G(T)/W_H(T)$.

Indeed, it follows from Prop. 7 of §2, no. 5, applied to $N_G$(H), that $N_G(H)/H$ is isomorphic to $W_{N(H)}(T)/W_H$(T), where $W_{N(H)}(T)$ is the set of elements of $W_G(T)$ whose representatives in $N_G(T)$ normalize H. Let $n\in N_G$(T), and let $w$ be its class in $W_G(T)$. By Chap. III, §9, no. 4, Prop. 11, $n$ normalizes H if and only if (Ad $n$)$(L(H)) = L(H)$; in view of Prop. 5 of no. 3, this also means that the subset $R(H,T)$ of R is stable under $w$, hence the corollary.