Exercise 2, § 2

Assume that $R$ is irreducible. Let $\{ \alpha_1, \ldots, \alpha_l \}$ be a basis of $R$, let $\tilde{\alpha} = \sum_i n_i \alpha_i$ be the highest root of $R$ and let $f$ be the connection index of $R$. Show that $f - 1$ is equal to the number of indices $i$ such that $n_i = 1$.