$L$ is a semisimple lie algebra and $H$ is the maximal toral subalgebra. Then the killing form $\kappa$ restricting to $H$ is non degenerated. Therefore we can associate $\varphi \in H^*$ with $t_\varphi \in H$ s.t. $\varphi(*) = \kappa (t_\varphi , *)$ .
For a root $\alpha \neq 0$, we have $t_\alpha$ such that $\kappa (t_\alpha , *) = \alpha$. Then how to show that $\kappa( t_\alpha , t_\alpha) \neq 0$?
A probably accessible way is like this. Since $t_\alpha \in H$, it is ad-semisimple. If we can show that $t_\alpha$ is ad-nilpotent. Then $\mathrm(ad )_L t_\alpha = 0$ and $t_\alpha \in Z(L)$ which contradicts the fact that $L$ is semisimple.