Definition 2

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

Lie Groups and Lie Algebras, chapter VIII, § 2. Root system of a split semi-simple Lie algebra

A root of $(\mathfrak{g},\mathfrak{h})$ is a non-zero weight of $\mathfrak{h}$ on $\mathfrak{g}$ .

Denote by $R(\mathfrak{g},\mathfrak{h})$ , or simply by R, the set of roots of $(\mathfrak{g},\mathfrak{h})$ . We have

\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha\in R}\mathfrak{g}^{\alpha}

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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