I have been reading di Francesco's Conformal Field Theory section on Lie algebras (chp 13, pg 491). Following their notation, let $[\cdot,\cdot]$ be the multiplication in the Lie algebra. The Cartan-Weyl basis first chooses the generators $\{H^i\}$ where $[H^i, H^j] = 0$, the maximal abelian subalgebra $H$ of the Lie algebra. The remaining generators $\{E^a\}$ are chosen such that $[H^i, E^a] = \alpha^i E^a$.
Now, they claim that because $H$ is the maximal abelian subgroup, the roots are non-degenerate. I have been trying to prove this claim to no avail. I have tried to assume the contrary and construct a new element of $H$ to contradict the maximality of $H$, but have been unsuccessful. How does one prove the statement that the roots $\alpha^i$ are non-degenerate?