Proposition 6

Tag 00U7, permanent. Cite it as [Bourbaki, Tag 00U7].

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

Let $\mathfrak{g}$ be a Lie algebra, $k'$ an extension of $k$ , and $\mathfrak{g}'=$ $\mathfrak{g}\otimes_kk'$ .

(i) An element $x$ of $\mathfrak{g}$ is regular in $\mathfrak{g}$ if and only if $x\otimes 1$ is regular in $\mathfrak{g}'$ .

(ii) rk( $\mathfrak{g}$ ) $=$ rk( $\mathfrak{g}'$ ).

This follows from Remark 2.

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

Theorem 2, § 3

Cites

Remark 2