I was reading non-linear realisation of a group on Wikipedia. On this page, it states: Given a Lie algebra $\mathfrak{g}$ and its Cartan subalgebra $\mathfrak{h}$, $\mathfrak{g}$ splits into $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{f}$ where $\mathfrak{f}$ is the supplement of the Cartan subalgebra such that $[\mathfrak{f},\mathfrak{f}]\subset\mathfrak{h}$ and $[\mathfrak{h},\mathfrak{f}]\subset\mathfrak{f}$.
While the second statement $[\mathfrak{h},\mathfrak{f}]\subset\mathfrak{f}$ is clearly true by definition of Cartan subalgebra as $\mathfrak{h}=${$g\in\mathfrak{g}|\exists n\in\mathbb{N}_0, ad(h)^ng=0\forall h\in\mathfrak{h}$}. I am not so certain about the first statement $[\mathfrak{f},\mathfrak{f}]\subset\mathfrak{h}$.
For example, let X and Y be two vectors in two different root spaces of the Lie algebra. If these two roots add up to become another root then the commutator of X,Y will be in the root space of the added root, clearly not in the Cartan subalgebra.
Could someone offer an explanation?