Corollary 2

Assume that $V$ is equipped with the structure of an ordered vector space such that, for this structure, every root of $R$ is either positive or negative. Let $P$ be the set of positive roots for this structure. Then there exists a unique chamber $C$ of $R$ such that $P = R_+(C)$.

Indeed, $P$ satisfies condition (ii) of Cor. 1.

This corollary applies in particular when the order being considered is total, the condition on $R$ then being automatically satisfied. Recall that such an order can be obtained, for example, by choosing a basis $(e_i)_{1 \leq i \leq n}$ of $V$ and taking the lexicographic order on $V$, so $x = \sum_i \xi_i e_i$ is $\geq 0$ if all the $\xi_i$ are 0, or if $\xi_i > 0$ for the smallest index $i$ such that $\xi_i \neq 0$.