The root poset

There are two different graphs shown below. The one labelled Root Poset has vertices as roots, with the simple roots at the bottom, and a labelled edge \alpha \xto{i} \beta if \alpha + \alpha_i = \beta. All edges in the root poset graph are directed edges, going up the page.

The one labelled Root reflections has vertices as roots, but this time there is an i-labelled undirected edge \set{\alpha, \beta} if s_i(\alpha) = \beta. The root reflection graph is disconnected in non-simply-laced types, separated into two pieces by the length of the roots.

My plan for this was for it to eventually become part of a how-to-build-a-root-system explanation, starting from a Cartan matrix.