Suppose you have a subset $\alpha_1,\dots,\alpha_r$ of simple roots in a reflection group whose corresponding reflections $\{s_1,\dots,s_r\}$ pairwise commute.
Why are the only positive roots expressible as a linear combination of these $\alpha_i$ the $\alpha_i$ themselves?
This fact comes up when counting the number of hyperplanes that intersect a particular plane in computing the order of a Coxeter element.