In a paper 1105.5073, the authors took a simply-laced root system $\Delta$ of type $G=A,D,E$, and then counted the number of unordered triples $(\alpha,\beta,\gamma)$ of roots which sum to zero: $\alpha+\beta+\gamma=0$.

They found that there are $rh(h-2)/3$ such triples, where $r$ is the rank of $G$ and $h$ is the Coxeter number of $G$.

Of course we can show this by studying the simply-laced root systems one-by-one (which is what the authors did.)

My question is if there is a nicer way to show this without resorting to a case-by-case analysis, using the general property of the Coxeter element, etc.

p.s.

Should I ask this at math.stackexchange.com?