Remark 3

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

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

Let $\mathfrak{g}$ be a semi-simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ , and $\rho$ a finite dimensional injective representation of $\mathfrak{g}$ such that $\rho (\mathfrak{h})$ is diagonalizable. Then ad $_{\mathfrak{g}}x$ is diagonalizable for all $x\in \mathfrak{h}$ (Chap. VII, §2, no. 1, Example 2), so $\mathfrak{h}$ is splitting.

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

Nothing.

Cites

Chap. VII, §2, no. 1, Example 2