Lemma 4

For all $w \in W$, the characteristic polynomial of $w$ has integer coefficients.

We know (no. 6, Th. 3) that $\{ \alpha_1, \ldots, \alpha_l \}$ is a basis of the subgroup $Q(R)$ of $V$ generated by $R$. Since $w$ leaves $Q(R)$ stable, its matrix with respect to $\{ \alpha_1, \ldots, \alpha_l \}$ has integer entries; hence its characteristic polynomial has integer coefficients.

Let $P$ be the characteristic polynomial of $c$. The above lemma shows that the coefficients of $P$ are integers. By Chap. V, § 6, no. 2, Cor. 2 of Prop. 3, the primitive $h$th root of unity $z = \exp(\frac{2i\pi}{h})$ is a simple root of $P$. Every conjugate of $z$ over $\mathbf{Q}$ is therefore also a simple root of $P$. But we know (Algebra, Chap. V) that the primitive $h$th roots of unity are conjugate over $\mathbf{Q}$. They are thus all simple roots of $P$, which proves the proposition.