Exercise 8, § 3

Tag 010M, permanent. Cite it as [Bourbaki, Tag 010M].

Lie Groups and Lie Algebras, VIII, 230

Choose a basis of R, and hence a set of positive roots $R_+$ . Assume that R is irreducible. Let $\widetilde{\alpha}$ be the highest root, S the set of roots orthogonal to $\widetilde{\alpha}$ , and $S_+= S\cap R_+$ .

a) Let

\mathfrak{g}'=\mathfrak{g}^S\oplus \mathfrak{h}_S,\mathfrak{m}_+=\sum_{\alpha\in S_+}\mathfrak{g}^{\alpha},\mathfrak{m}_-=\sum_{\alpha\in S_+}\mathfrak{g}^{-\alpha}

Then $\mathfrak{g}'$ is semi-simple and $\mathfrak{g}'=\mathfrak{h}_S\oplus \mathfrak{m}_+\oplus \mathfrak{m}_-$ .

b) Put:

\mathfrak{n}_+=\sum\mathfrak{g}^{\alpha},\mathfrak{p}=\sum\mathfrak{g}^{\alpha},\mathfrak{p}_0=\sum\mathfrak{g}^{\alpha}

\alpha \in R_+\alpha \in R_+-S_+\alpha \in R_+-S_+-\{\widetilde{\alpha}\}

Then

\mathfrak{n}_+=\mathfrak{m}_+\oplus \mathfrak{p},[\mathfrak{m}_+,\mathfrak{p}_0]\subset \mathfrak{p}_0,[\mathfrak{n}_+,\mathfrak{g}^{\widetilde{\alpha}}] = 0

In particular, $\mathfrak{p}$ is an ideal of $\mathfrak{n}_+$ .

c) For all $\alpha \in R_+$ -- $S_+$ -- $\{\widetilde{\alpha}\}$ , there exists a unique $\alpha '\in R_+$ -- $S_+$ -- $\{\widetilde{\alpha}\}$ such that $\alpha +\alpha '= \widetilde{\alpha}$ . For all $\alpha \in R_+$ -- $S_+$ -- $\{\widetilde{\alpha}\}$ , a non-zero element $X_{\alpha}$ of $\mathfrak{g}^{\alpha}$ can be chosen so that

$[X_{\alpha}, X_{\alpha'}] =\pm X_{\widetilde{\alpha}}$ if $\alpha \in R_+-S_+- \{\widetilde{\alpha}\}$

$[X_{\alpha}, X_{\beta}] = 0$ if $\alpha , \beta \in R_+-S_+, \beta \not=\alpha '.^{11}$

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