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 $\dim \mathfrak g_{\alpha}=1$?