Decomposition of set of roots for a Lie algebra and its Cartan subalgebra Consider a finite dimensional complex semi-simple Lie algebra $L$ with Cartan subalgebra $H$ (i.e. every $h\in H$ is $ad$-nilpotent). Denote $\Phi=\Phi(L,H)$ the set of roots.
Assume $\Phi=\Phi_1\cup\Phi_2$ for non empty $\Phi_i$ and $(\alpha,\beta)=0$ for all $\alpha\in\Phi_1,\beta\in\Phi_2$. Here $(\alpha,\beta)=\kappa(H^\alpha,H^\beta)$, where $\kappa$ is the Killing form and $H^\alpha$ is the unique element such that $\kappa (H^\alpha,\cdot)=\alpha$ (analogue for $H^\beta$). Set, for $i=1,2$, $$L_i=\text{span}_{\mathbb{C}}\{H^{\alpha}:\alpha\in\Phi_i\}\oplus\bigoplus_{\alpha\in\Phi_i}L_\alpha.$$
Why do we have $L=L_1\oplus L_2$ as Lie algebras?

It is obvious that $L=L_1\oplus  L_2$ as vector spaces. To check that it holds as Lie algebras, you need to show that $[L_1,L_2]=0$. But this follows immediately from the orthogonality assumption, and the fact that $\alpha+\beta\notin\Phi$ for all $\alpha\in\Phi_1$ and $\beta\in\Phi_2$.
