Exercise 2, § 1

Let R be a root system, $\alpha$ and $\beta$ two roots. If t is a scalar such that $\beta + t\alpha \in R$, then $2t \in \mathbf{Z}$. (For $n(\beta + t\alpha, \alpha) = n(\beta, \alpha) + 2t.$) If $\alpha$ is indivisible, then $t \in \mathbf{Z}$. (If not, show, by using Prop. 9, that there exists a root $\gamma$ orthogonal to $\alpha$ such that $\gamma + \frac{1}{2}\alpha \in R$: then $(\gamma + \frac{1}{2}\alpha|\alpha) > 0$, so $\frac{1}{2}\alpha \in R.$)