Proposition 8

If $W$ is finite, the graph of $(W, S)$ is a forest (Chap. IV, Appendix).

If not, this graph would contain a circuit $(s_1, \ldots, s_n)$, $n \geq 3$. Putting $m_i = m(s_1, s_{i+1})$, $1 \leq i < n$, and $m_n = m(s_n, s_1)$, this means that $m_i \geq 3$ for all $i$. Let

Then $B_M(x, x) = n + 2 \sum_{i<j} B_M(e_{s_i}, e_{s_j})$. Now$^2$ and similarly for $B_M(e_{s_n}, e_{s_1})$. Since the other terms in the sum are $\leq 0$, we obtain contrary to the fact that $B_M$ is positive non-degenerate.