Corollary

Tag 00YT, permanent. Cite it as [Bourbaki, Tag 00YT].

Lie Groups and Lie Algebras, chapter VIII, § 2. Root system of a split semi-simple Lie algebra

Let $\Phi$ be the Killing form of $\mathfrak{g}$ . For all $a, b\in \mathfrak{h}$ ,

\Phi (a, b) =\sum_{\gamma\in R}\gamma (a)\gamma (b)

Indeed, ad $a$ .ad $b$ leaves each $\mathfrak{g}^{\gamma}$ stable, and its restriction to $\mathfrak{g}^{\gamma}$ is the homothety with ratio $\gamma (a)\gamma (b)$ ; if $\gamma \not= 0$ , dim $\mathfrak{g}^{\gamma}= 1$ .

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

Remark 2, § 2
Proposition 2, § 2

Cites

Nothing.