In Humphreys book in "Reflection Groups and Coxeter Groups" he defines a root system $\Phi$ to be crystallographic if it satisfies $\frac{2(\alpha, \beta)}{(\beta, \beta)} \in \mathbb{Z}$ $(\star)$ for all $\alpha, \beta \in \Phi$ and he states that it is enough to require that the ratios be integers when $\alpha, \beta \in \Delta$, where $\Delta$ is the simple system of a Coxeter group (the elements of $\Phi$ are either non-negative or either non-positive linear combinations of the elements of $\Delta$ and $\Delta$ is a basis for the vector space where the Coxeter group acts).
Having the result $(\star)$ for the elements on $\Delta$, I don't see how to show this holds for the elements of $\Phi$.
Thank you in advance
By the assumption on simple roots, we know that if $\alpha, \beta \in \Delta$, then $s_\alpha(\beta)$ is an integral linear combination of $\alpha$ and $\beta$. Since $\{s_\alpha: \alpha \in \Delta\}$ generates $W$, then for any $w \in W$, $\beta \in \Delta$, we know that $w \beta$ is an integral linear combination of elements of $\Delta$. But every root in $\Phi$ is of the form $w \beta$ (Corollary 1.5) and is thus an integral linear combination of elements of $\Delta$.
Now the rest is easy. To show that $f(\alpha,\beta) := \frac{2(\alpha, \beta)}{(\beta,\beta)} \in \mathbb{Z}$ for arbitrary $\alpha, \beta \in \Phi$, first note that $f(\alpha,\beta)$ is invariant under $W$, so by replacing $(\alpha,\beta)$ by $(w\alpha, w\beta)$ for a suitable $w \in W$, we may assume that $\beta \in \Delta$. Then note that $f(\alpha, \beta)$ is linear in $\alpha$, so we may assume $\alpha \in \Delta$ as well, and we are done by our original assumption that $f(\alpha,\beta) \in \mathbb{Z}$ for $\alpha, \beta \in \Delta$.