I am following Fuchs book on Lie algebras. Some topics are presented in a not so clear way, but it seems that he claims the following:
Claim: Writing $\text{span}_{\mathbb C}\{H^\alpha,E^{\pm \alpha}\} =: \mathfrak{sl}(2)_\alpha$, we have \begin{equation} \mathfrak g = \bigoplus_{\alpha\in\Phi^+} \mathfrak {sl}(2)_\alpha \end{equation} However, in section 13.2, he says that in order to study the representations of a semisimple Lie algebra, we only have to worry about the $\mathfrak{sl}(2)_{\alpha}$ with $\alpha$ a simple root. But of course we cannot ignore the non-simple positive roots in the decompostion above. So what's going on?
EDIT: This may help, although I am not sure how: From section 7.1 we can say that the semisimple Lie algebra is "algebraically generated" by the Cartan subalgebra generators and the ladder operators corresponding to simple roots. This "algebraic generation" means that we can not only use linear combinations but also the Lie brackets.
I'd like to expand my comment a little bit. The fact that restriction of finite dimensional irreducible repr. $V$ to Cartan subalgebra $\mathfrak{h}$ is much more intuitively clear from Lie Group perspective:
Let $G$ be compact connected Lie group, corresponding to semisimple Lie algebra $\mathfrak{g}$. Here instead of Cartan we have some maximal compact torus $T$, isomorphic to $\mathrm{U}(1)^{\ell}$, where $\ell = \dim \mathfrak{h}$. Character corresponding to $V$ is a function on conjugacy classes (this was true in case of finite discrete $G$, and the analogy is much deeper). It turns out that all conjugacy classes intersect $T$ nontrivially, so restriction to $T$ really determines the character of $V$, which determines $V$ completely.
Moreover Weyl group $\mathcal{W}$ measures exactly how nontrivial the intersection is:
$\mathrm{Theorem:}$ If $g_1, g_2$ are elements in $T$ which are in the same conjugacy class in $G$, then there is $w\in \mathcal{W}$ such that $w.g_1 = g_2$.
The reason why all that magic with Cartans and simple roots seems unmotivated is that the subject is so well developed and understood that it takes a few chapters in books to fully classify semisimple Lie algebras with description of their representations.