Let $\mathfrak{h}$ the Cartan subalgebra of a simply-laced (ADE type) Lie algebra of rank $r$, $\mathfrak{g}$, and $\Phi$ the associated root system. Furthermore, let $\Phi^+$, $\Pi$, denote the simple and positive roots, respectively. We then have a Cartan-Weyl basis $\mathfrak{g} = \mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\text{span}(E_\alpha)$.
Now pick an element $H\in\mathfrak{h}$ so that: $$ [H,E_\alpha] = \alpha(H) E_\alpha\,,\qquad \forall\,\alpha\in\Phi $$ For the simple roots, we define $$ \alpha_i(H) = w_i,\qquad i = 1,...,r\,,\quad \alpha_i\in \Pi $$ For concreteness, I have in mind nilpotent orbits where $H$ is the Cartan element of the Jacobson-Morozov triplet, and $w_i$ are the labels of the weighted Dynkin diagrams, but the question is more general.
I am interested in the quantity $$ F = \sum_{\alpha\in\Phi^+}\alpha(H)^3 $$ expressed in terms of the value of the simple roots, $w_i$. I have found that one lower exponents, the expressions are very simple: $$ \sum_{\alpha\in\Phi^+}\alpha(H) = 2 \rho(H) = 2\sum_{i=1}^r \rho^i w_i\\ \sum_{\alpha\in\Phi^+}\alpha(H)^2 =\frac{1}{2}\kappa(H,H) \propto \sum_{i,j=1}^r w_i A^{-1}_{ij} w_j $$ where $\rho^i$ are the coefficients of the Weyl vector, $\kappa$ the Killing form, and $A^{-1}$ the inverse Cartan matrix (the second one requires to pick a basis for $\mathfrak{h}$ and find $H$ given $w_i$).
Is there a simple way of expressing $F$ in terms of the $w_i$ and some "natural" quantities like the Cartan Matrix or the Weyl vector, whose numerical values are known (from e.g. Bourbaki's "Planches" for the Chevalley basis)? If so, is there similar expression for arbitrary exponents?
It's easy to see that there's a three-tensor $c_{ijk}$ such that $F = \sum_{i,j,k=1}^r c_{ijk}w_iw_jw_k$, but I couldn't find a general relation.