Corollary 3

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

Lie Groups and Lie Algebras, chapter IX, § 4. Root system associated to a compact group

The homomorphism $\pi_1(D(G))\rightarrow \pi_1(G)$ corresponding to the inclusion of D(G) into G induces an isomorphism from $\pi_1(D(G))$ to the torsion subgroup of $\pi_1(G)$ .

Indeed, $T\cap D(G)$ is a maximal torus of D(G) (§2, no. 3, Prop. $1c$ ) $)$ ; from the exact sequence

0\longrightarrow \Gamma (T\cap D(G))\longrightarrow \Gamma (T)\longrightarrow \Gamma (T/(T\cap D(G)))\longrightarrow 0

and Proposition 11, we obtain an exact sequence

0\longrightarrow \pi_1(D(G))\longrightarrow \pi_1(G)\longrightarrow \Gamma (T/(T\cap D(G)))\longrightarrow 0

hence the corollary, since $\pi_1(D(G))$ is finite and $\Gamma (T/(T\cap D(G)))$ is free.

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 1, § 5

Cites

§2, no. 3
Proposition 11