Let $\mathfrak g$ be a semi-simple Lie algebra with Cartan subalgebra $\mathfrak h$ and root system $\Phi$. I have seen it stated that for any $\alpha \in \Phi$, $-\alpha \in \Phi$, and that this is a consequence of the antisymmetry of $\mathsf {ad} : \mathfrak h \to \mathsf {End}(\mathfrak h)$ with respect to the Killing form $k$. I have been unable to prove this. I have tried to prove it by expanding
$$k(\mathsf{ad} _h(x), y) = k([h, x],y) = -k(x, [h,y]),$$
but I haven't been able to get anywhere. Do you have any hints, or the full proof?