$\newcommand{\g}{\mathfrak{g}}$ I have a finite question:
Let $\g$ be a finite dimensional semisimple lie algebra and let a choice of a weyl chamber be made. If $\gamma>0$ is a particular positive root, then is the set $R_\gamma$ of positive roots that can't be written as $\gamma+\alpha$ for some $\alpha>0$ just $\gamma$ and all of the simple roots?
Reason for asking:
In the case when $\g$ is a Kac-Moody lie algebra of a given cartan matrix, I am trying to see what the expression $\sum_{i, \alpha \in R_\gamma} [e_{-\alpha}^i,x]e_\alpha^i$ is, where I am taking a sum over products of dual basis elements of root spaces.
More motivation(added 3/10 12:57 pm): I calculated that this sum is one part of the casimir commutated with an element $x=e_\gamma^i$. Namely $[\sum_{\alpha>0} e_{-\alpha}^ie_{\alpha}^i ,x]=\sum_{i, \alpha \in R_\gamma} [e_{-\alpha}^i,x]e_\alpha^i$