= Solution
For the same graph, let $e_0$ correspond to the central vertex and $e_1,\ldots,e_4$ to the leaves. The associated simply-laced <Coxeter Gram matrix> has
$$
(e_i,e_i)=2,qquad(e_0,e_j)=-1,qquad(e_i,e_j)=0
$$
for distinct leaves. The nonzero vector
$$
\delta=2e_0+e_1+e_2+e_3+e_4
$$
satisfies $(\delta,e_i)=0$ for every basis vector. Thus the form is degenerate; in fact it is positive semidefinite with one-dimensional radical, as expected for the affine graph $\widetilde D_4$.
The geometric form of a finite Coxeter group is positive definite. Since this form is degenerate, the group cannot be finite.
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