Proposition 7

Let $H$ be a closed subgroup of $\mathrm{G}$ containing $\mathrm{T}$.

a) Denote by $\mathrm{W}_{H}(\mathrm{T})$ the subgroup $\mathrm{N}_{H}(\mathrm{T})/\mathrm{T}$ of $\mathrm{W}_{G}(\mathrm{T})$; the group $H/H_{0}$ is isomorphic to the quotient group $\mathrm{W}_{H}(\mathrm{T})/\mathrm{W}_{H_{0}}(\mathrm{T})$.

b) $H$ is connected if and only if every element of $\mathrm{W}_{G}(\mathrm{T})$ that has a representative in $H$ belongs to $\mathrm{W}_{H_{0}}(\mathrm{T})$.

Assertion a) follows from Cor. 8 of Th. 2 (no. 2), and assertion b) is a particular case of a).