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.



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> Is it true that $\dim \mathfrak g\_{\alpha}=1$? 
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Dimension of restricted root spaces of split Lie algebras
Yes. See Helgason (Prop. 6.3, [p. 430](https://books.google.com/books?id=a9KFAwAAQBAJ&pg=PA430); 6(a), [p. 531](https://books.google.com/books?id=a9KFAwAAQBAJ&pg=PA531)) or Onishchik–Vinberg [1990](https://mathscinet.ams.org/mathscinet-getitem?mr=1064110) (23–25, [p. 274](https://books.google.com/books?id=TV7sCAAAQBAJ&pg=PA274)).


