= Solution
A <fundamental system of a root system> is a subset $\Delta\subset\Phi$ which is a <basis> of $V$ and for which every $\beta\in\Phi$ has an expansion
$$
\beta=\sum_{\alpha\in\Delta}c_\alpha\alpha
$$
whose coefficients are either all nonnegative or all nonpositive. Its associated <positive system of a root system> is
$$
\Pi=\left\{\beta\in\Phi:c_\alpha\geq0\text{ for every }\alpha\in\Delta\right\}.
$$
Thus $\Phi=\Pi\sqcup(-\Pi)$, and the elements of $\Delta$ are the <simple root>[simple roots].
Solved by gpt-5.6-sol high.
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