Remark

Tag 01B2, permanent. Cite it as [Bourbaki, Tag 01B2].

Lie Groups and Lie Algebras, chapter IX, § 3. Compact forms of complex semi-simple Lie algebras

With the preceding notations, we have the following formulas:

[h, u_{\alpha}] =-i\alpha (h)v_{\alpha},[h, v_{\alpha}] =i\alpha (h)u_{\alpha},[u_{\alpha}, v_{\alpha}] = 2iH_{\alpha},(h\in \mathfrak{h}) \tag{9}

[u_{\alpha}, u_{\beta}] = N_{\alpha ,\beta}u_{\alpha+\beta}+ N_{\alpha ,-\beta}u_{\alpha-\beta},\alpha \not=\pm \beta \tag{10}

[v_{\alpha}, v_{\beta}] =-N_{\alpha ,\beta}u_{\alpha+\beta}+ N_{\alpha ,-\beta}u_{\alpha-\beta},\alpha \not=\pm \beta \tag{11}

[u_{\alpha}, v_{\beta}] = N_{\alpha ,\beta}v_{\alpha+\beta}-N_{\alpha ,-\beta}v_{\alpha-\beta},\alpha \not=\pm \beta \tag{12}

(in the last three formulas, it is understood, as usual, that $N_{\gamma ,\delta}= 0$ if $\gamma +\delta$ is not a root).

Note that $\sum\mathbf{R}u_{\alpha}$ is a real subalgebra of $\mathfrak{a}$ , namely $\mathfrak{a}_0\cap \mathfrak{a}_u$ .

Let Q(R) be the group of radical weights of R (Chap. VI, §1, no. 9). Recall that to any homomorphism $\gamma : Q(R)\rightarrow \mathbf{C}^*$ is associated an elementary automorphism $f(\gamma )$ of $\mathfrak{a}$ such that $f(\gamma )(h) =h$ for all $h\in \mathfrak{h}$ and $f(\gamma )X_{\alpha}=$ $\gamma (\alpha )X_{\alpha}$ (Chap. VIII, §5, no. 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.

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