Lemma 1

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

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

Let V be a vector subspace of $\mathfrak{g}$ and R(V) the set of $\alpha \in R$ such that $\mathfrak{g}^{\alpha}\subset V$ . Then, $(V\cap \mathfrak{h}) +\sum_{\alpha\in R(V)}\mathfrak{g}^{\alpha}$ is the largest vector subspace of V stable under ad $\mathfrak{h}$ .

A vector subspace W of V is stable under ad $\mathfrak{h}$ if and only if

W = (W\cap \mathfrak{h}) +\sum_{\alpha\in R}(W\cap \mathfrak{g}^{\alpha})

(Algebra, Chap. VII, §2, no. 2, Cor. 1 of Th. 1). The largest vector subspace of V stable under ad $\mathfrak{h}$ is thus $(V\cap \mathfrak{h}) +\sum_{\alpha\in R}(V\cap \mathfrak{g}^{\alpha})$ . But $V\cap \mathfrak{g}^{\alpha}=\mathfrak{g}^{\alpha}$ for $\alpha \in R(V)$ , and $V\cap \mathfrak{g}^{\alpha}= 0$ for $\alpha \notin R(V)$ since dim $\mathfrak{g}^{\alpha}= 1$ . Q.E.D.

For any subset P of R, put

\mathfrak{g}^P=\sum_{\alpha\in P}\mathfrak{g}^{\alpha}\mathfrak{h}_P=\sum_{\alpha\in P}\mathfrak{h}_{\alpha}

If $P\subset R$ and $Q\subset R$ , we clearly have

[\mathfrak{h},\mathfrak{g}^P]\subset \mathfrak{g}^P \tag{1}

[\mathfrak{g}^P,\mathfrak{g}^Q] =\mathfrak{g}^{(P+Q)\cap R}+\mathfrak{h}_{P\cap(-Q)} \tag{2}

Recall (Chap. VI, §1, no. 7, Def. 4) that a subset P of R is said to be closed if the conditions $\alpha \in P, \beta \in P, \alpha +\beta \in R$ imply $\alpha +\beta \in P$ , in other words if $(P + P)\cap R\subset P$ .

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

Proposition 3, § 3

Cites

Algebra, Chap. VII, §2, no. 2, Cor. 1 of Th. 1 not in this corpus
Chap. VI, §1, no. 7, Def. 4 not in this corpus