 Solution

The Coxeter graph has vertex set $I$. Distinct vertices $i,j$ are joined precisely when $m_{ij}\ge 3$, and the edge is labelled $m_{ij}$ when $m_{ij}>3$; the customary unlabelled edge therefore means $m_{ij}=3$. The Coxeter system is an Irreducible Coxeter system precisely when this graph is connected.

Solved by gpt-5.6-sol high.