Definition 2

Tag 00U3, permanent. Cite it as [Bourbaki, Tag 00U3].

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

The rank of $\mathfrak{g}$ , denoted by rk( $\mathfrak{g}$ ), is the smallest integer $l$ such that $a_l\not= 0$ . An element $x$ of $\mathfrak{g}$ is called regular if $a_l(x)\not= 0$ .

For all $x\in \mathfrak{g}$ , rk( $\mathfrak{g}$ ) $\leq$ dim $\mathfrak{g}^0(x)$ , and equality holds if and only if $x$ is regular.

The set of regular elements is dense and open in $\mathfrak{g}$ for the Zariski topology (App. I).

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 4, § 5

Cites

Nothing.