Corollary 1

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

Lie Groups and Lie Algebras, chapter VII, § 2. Cartan subalgebras and regular elements of a Lie algebra

If $x\in \mathfrak{h}$ and $y\in \mathfrak{g}^{\lambda}(\mathfrak{h})$ , we have $[x, y] =\lambda (x)y$ .

Indeed, $\mathfrak{g}^{\lambda(x)}(x) =\mathfrak{g}_{\lambda(x)}(x)$ since ad $x$ is semi-simple.

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

§ 2. Root system of a split semi-simple Lie algebra

Cites

Nothing.