Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 3 / c

By part (a), every full subdiagram gives a standard parabolic Coxeter subgroup. In the $B_{n+1}$ diagram, the $n$ vertices away from the endpoint incident to the edge labelled four form an $A_n$ chain. Thus

The $E_8$ diagram is a trivalent tree whose three arms have lengths $1,2,4$ beyond the central vertex. Removing the endpoint of the arm of length one leaves a chain of seven vertices, hence an $A_7$ subdiagram. Removing the outer endpoint of the arm of length two leaves arms of lengths $1,1,4$, the $D_7$ diagram. Consequently