Exercise 4, § 4

Assume that $\mathrm{Card}(S) = 3$. If $s \in S$, let $a(s) = m(u, v)$, where $\{u, v\} = S - \{s\}$. Let $A = \sum_{s \in S} 1/a(s)$.

a) If $A > 1$, show that $B_M$ is non-degenerate and positive (in which case $W$ is finite).

b) If $A = 1$, show that $B_M$ is degenerate and positive.

c) If $A < 1$, show that $B_M$ is non-degenerate and of signature $(2, 1)$ (cf. Algebra, Chap. IX, § 7, no. 2).

Show that, in case a), the order $q$ of $W$ is given by the formula $q = 4/(A - 1)$ (use Exerc. 5 of § 3).