Suppose that $\mathfrak{g}$ is a semisimple Lie algebra, with root spaces $\mathfrak{g}^{\alpha}$ for $\alpha \in R$ where $R$ is the associated (reduced) root system for $\mathfrak{g}$. In Humphreys (p. 39), part $(f)$ of Proposition 8.4 says:
$\mathfrak{g}$ is generated (as a Lie algebra) by the root spaces $\mathfrak{g}^{\alpha}$.
I am unclear as to what the meaning of "generated" is, in this context. Does the content of this statement essentially have the same meaning as the content of the theorem about the Cartan decomposition of $\mathfrak{g}$, or are we saying more/something different here?