 Positive roots for a chain Cartan matrix

Suppose the Cartan matrix has diagonal two, entries minus one between adjacent indices, and all other entries zero. Normalize the simple-root squared lengths to two. Every root is a Weyl group translate of a simple root, so all roots have squared length two. If a positive root has coefficients $c_1,\dots ,c_l$ in the simple-root basis, let $c_0=c_{l+1}=0$. ThenThe nonnegative integer coefficients force exactly one upward jump of one and one downward jump of one. Thus the coefficients are one on a consecutive interval and zero elsewhere. Conversely, an interval root can be extended by the next simple root: its Cartan pairing with that root is minus one, so the corresponding simple reflection adds it. Starting from a simple root constructs every interval and proves the displayed description. This also identifies the root system as type $A_l$ without needing to enumerate reflection words individually.