Let $\mathfrak{g}$ be a simple Lie algebra with Cartan matrix $A$, $h^i, e^i = E^{\alpha_i}, f^i = E^{- \alpha_i}$ form the Chevalley basis for the simple roots $\alpha_i$ (and simple coroots $\alpha_i^\vee$), with the standard commutation relation $$ [h^i, E^{\alpha_j}] = A_{ji}E^{\alpha_j}, \quad [h^i, E^{ - \alpha_j}] = - A_{ji}E^{ - \alpha_j}, \quad [E^{\alpha_i}, E^{- \alpha_i}] = h^i \ . $$
The Killing form is Chosen such that $$ K(h^i, h^j) = (\alpha_i^\vee, \alpha_j^\vee) \ , \qquad K(E^\alpha, h^i) = 0, \qquad K(E^{\alpha}, E^{\beta}) \propto \delta_{\alpha+\beta , 0} \ , $$ where the inner product $(\mu, \nu)$ is the standard one such that the long roots have norm-squared 2.
Suppose $K(E^{\alpha_i}, K^{- \alpha_i})$ is also known. Is there a systematic way to determine all $K(E^\alpha, E^{- \alpha})$?
(By some direct checks using explicit matrix representation, it seems that for simply-laced algebras, $K(E^{\alpha_i}, E^{-\alpha_i}) = 1$, and $K(E^{\alpha}, E^{-\alpha}) = \pm 1$.)