= Solution
The <reflection group of a root system> is
$$
W(\Phi)=\langle s_\alpha:\alpha\in\Phi\rangle\leq O(V).
$$
Every generating reflection permutes $\Phi$, so every $w\in W(\Phi)$ does too. If $\Delta$ is fundamental, then $w\Delta$ is a basis contained in $\Phi$. Writing
$$
\beta=\sum_{\alpha\in\Delta}c_\alpha\alpha
$$
shows that
$$
w\beta=\sum_{\alpha\in\Delta}c_\alpha w\alpha;
$$
the coefficients are unchanged and therefore still have one sign. Thus $w\Delta$ is another fundamental system, and $W(\Phi)$ acts on the set of all fundamental systems.
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