Proposition 12

Tag 0109, permanent. Cite it as [Bourbaki, Tag 0109].

Lie Groups and Lie Algebras, chapter VIII, § 3. Subalgebras of split semi-simple Lie algebras

Let $\mathscr{H}$ be the set of hyperplanes of $\mathfrak{h}_{\mathbf{R}}$ consisting of the kernels of the roots in R. Let $\mathscr{F}$ be the set of facets of $\mathfrak{h}_{\mathbf{R}}$ relative to $\mathscr{H}$ . Let $\mathscr{S}$ be the set of parabolic subalgebras of $(\mathfrak{g},\mathfrak{h})$ . For every $\mathfrak{p}=\mathfrak{h}+\mathfrak{g}^P\in \mathscr{S}$ , let $F(\mathfrak{p})$ be the facet associated to P. Then $\mathfrak{p} \rightarrow F(\mathfrak{p})$ is a bijection from $\mathscr{S}$ to $\mathscr{F}$ . If $\mathfrak{p}_1,\mathfrak{p}_2\in \mathscr{P}$ ,

\mathfrak{p}_1\supset \mathfrak{p}_2\Leftarrow \Rightarrow F(\mathfrak{p}_1)\subset \overline{F(\mathfrak{p}_2)}

This follows immediately from Lemma 3.

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

Lemma 3