I was reading Fulton and Harris' book when I found the following theorem:
In the proof of (ii) they say that $[X, Y] \in \mathfrak{h}$. Why is it true?
(It is an obvious corollary of this statement:
If $X$ is an element of a positive root space and $Y$ is an element of a negative root space then $[X, Y]$ is in the Cartan subalgebra.
Do they mean that it's true?)
UPD: Before this they're proving special cases of the proposition I've claimed (for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$) without saying any words about the general case.
Because if $\alpha$ and $\beta$ are roots, then $[\mathfrak g_\alpha,\mathfrak g_\beta]\subset\mathfrak g_{\alpha+\beta}$. In particular,$$[\mathfrak g_\alpha,\mathfrak g_{-\alpha}]\subset\mathfrak g_0=\mathfrak h.$$