Definition 4

Tag 0AVL, permanent. Cite it as [Bourbaki, Tag 0AVL].

Lie Groups and Lie Algebras, chapter VI, § 1. Root systems

Let $P$ be a subset of $\mathbf{R}$ .

(i) $P$ is said to be closed if the conditions $\alpha \in P, \beta \in P, \alpha + \beta \in \mathbf{R}$ imply $\alpha + \beta \in P$ .

(ii) $P$ is said to be parabolic if $P$ is closed and if $P \cup (-P) = \mathbf{R}$ .

(iii) $P$ is said to be symmetric if $P = -P$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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