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Let $\mathfrak g$ be a real simple split Lie algebra. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the Cartan decomposition. Let $\mathfrak a\subseteq \mathfrak p$ be a maximal abelian subalgebra. Let $\alpha \in \mathfrak a^*$ be a (restricted) root and let $\mathfrak g_{\alpha}$ be its root space.

Is it true that $\dim \mathfrak g_{\alpha}=1$?

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Yes. See Helgason (Prop. 6.3, p. 430; 6(a), p. 531) or Onishchik–Vinberg 1990 (23–25, p. 274).

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