= Solution
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
$$
W(A_n)\subset W(B_{n+1}).
$$
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
$$
\boxed{W(A_7)\subset W(E_8),
\qquad W(D_7)\subset W(E_8).}
$$
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