= Solution
A <root system> in the real <inner product> space $V$ is a finite spanning set $\Phi\subset V\setminus\{0\}$ such that
$$
\Phi\cap\mathbb R\alpha=\{\alpha,-\alpha\}
\quad\text{and}\quad
s_\alpha(\Phi)=\Phi
\qquad(\alpha\in\Phi),
$$
where the <orthogonal reflection>
$$
s_\alpha(v)=v-2\frac{(v,\alpha)}{(\alpha,\alpha)}\alpha.
$$
For a crystallographic root system one additionally requires $2(\beta,\alpha)/(\alpha,\alpha)\in\mathbb Z$; that condition is not needed for general finite reflection groups.
A <fundamental system of a root system> is a basis $\Delta\subset\Phi$ such that every root has either all nonnegative or all nonpositive coordinates in this basis. Its associated <positive system of a root system> is
$$
\Pi=\Phi\cap\operatorname{span}_{\mathbb R_{\geq0}}\Delta,
$$
and $\Phi=\Pi\sqcup(-\Pi)$.
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