Let $L_1,L_2$ be finite-dimensional semi-simple complex Lie algebras and let $H_1\subseteq L_1$, $H_2\subseteq L_2$ be Cartan subalgebras. Set $H=H_1\oplus H_2$ and $L=L_1\oplus L_2$. How can I show that $\Phi(L,H)=\Phi_1\cup \Phi_2$ where $\Phi_1:=\{\alpha\oplus 0\mid \alpha \in \Phi(L_1,H_1)\}$ and $\Phi_2:=\{0\oplus \beta\mid \beta \in \Phi(L_2,H_2)\}$?
($\Phi(L,H)$ denotes the root system of $L$ with respect to $H$)