Exercise 14, § 4

With the notation in no. 7, show that the automorphism of $\mathbf{R}^{l+1}$ which transforms $\varepsilon_1$ to $\varepsilon_2$, $\varepsilon_2$ to $\varepsilon_3$, ..., $\varepsilon_l$ to $\varepsilon_{l+1}$ and $\varepsilon_{l+1}$ to $\varepsilon_1$ induces a Coxeter transformation of the system R of type $A_l$.