Corollary

Tag 01JD, permanent. Cite it as [Bourbaki, Tag 01JD].

Lie Groups and Lie Algebras, chapter IX, § 2. Maximal tori of compact Lie groups

The canonical injection of $\mathfrak{t}$ into $\mathfrak{g}$ defines by passage to the quotient a homomorphism from $\mathfrak{t}/\mathrm{W}_{G}(\mathrm{T})$ to $\mathfrak{g}/\mathrm{Ad}(\mathrm{G})$ .

Denote this map by $j$ ; it is bijective and continuous (Prop. 6). We have a commutative diagram

\begin{array}{ccc} \mathfrak{t} & \xrightarrow{\ i\ } & \mathfrak{g} \\ {\scriptstyle p}\downarrow\phantom{{\scriptstyle p}} & & \phantom{{\scriptstyle q}}\downarrow{\scriptstyle q} \\ \mathfrak{t}/\mathrm{W}_{G}(\mathrm{T}) & \xrightarrow{\ j\ } & \mathfrak{g}/\mathrm{Ad}(\mathrm{G}) \end{array}

where $p$ and $q$ are quotient maps, and $i$ is the canonical injection. Since $i$ and $q$ are proper (General Topology, Chap. I, §10, no. 1, Prop. 2 and General Topology, Chap. III, §4, no. 1, Prop. 2 c)) and since $p$ is surjective, it follows that $j$ is proper (General Topology, Chap. I, §10, no. 1, Prop. 5), and hence is a homeomorphism.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Corollary 2, § 5

Cites

Prop. 6
*General Topology*, Chap. I, §10, no. 1, Prop. 2 not in this corpus
*General Topology*, Chap. I, §10, no. 1, Prop. 5 not in this corpus
*General Topology*, Chap. III, §4, no. 1, Prop. 2 not in this corpus