Let $R \subset V$ be a reduced root system, and $R' \subset R$. Assume that:
(i) $\alpha \in R' \ \to \ - \alpha \notin R'$,
(ii) $ \alpha, \beta \in R'$ and $\alpha + \beta \in R$ implies $\alpha + \beta \in R'$.
How can I show that there exists a base $B$ of $R$ s.t. $R'$ is contained in the set of positive roots wrt $B$?
I tried looking at the proof of base existence, but did not get any useful ideas from it.