Lemma 3

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

Lie Groups and Lie Algebras, chapter VIII, § 4. Split semi-simple Lie algebra defined by a reduced root system

The canonical images in $\mathfrak{a}$ of the elements $x_{\alpha}, h_{\beta}, x_{-\gamma}$ , where $\alpha , \beta , \gamma \in B$ , are linearly independent.

In the following, we identify $x_{\alpha}, h_{\alpha}, x_{-\alpha}$ with their canonical images in $\mathfrak{a}$ .

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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