Let $a,b$ be roots in a root system of a finite dimensional complex semisimple Lie algebra. I want to determine the possible values of $b(h_a)-a(h_b)$.
The difference equals zero when $a=b$ (clearly) or $a=-b$ (that is $a+b=0$) since $b(h_a)-a(h_b)=b[e_a,e_b]-a[e_b,e_a]=(b+a)[e_a,e_b]=0$ by how the Cartan Weyl basis acts under the Lie bracket.
I'm having trouble finding other cases. I know that $h_a=\frac{2}{\langle a,a\rangle}t_a$ where $t_a$ is the image under the map $\mathfrak{h}^*\rightarrow \mathfrak{h}$ with $\mathfrak{h}$ being a fixed Cartan subalgebra. However this doesn't get me much further.