Base of a root system 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. 
 A: Perhaps there's what you're looking for here : http://www.auburn.edu/~huanghu/math7360/Lie%20Algebra-3.2.pdf
The author uses a Lemma 3.3 at some point, which is shown here : http://www.auburn.edu/~huanghu/math7360/Lie%20Algebra-3.1.pdf

The idea : define a regular vector a vector which all scalar products with roots is non-zero. Given a regular vector $\gamma$, define $P$ the set of positive roots $\alpha$ such that $\langle\alpha|\gamma\rangle>0$. At last, define $\Delta=\lbrace\alpha\in P\;/\;\forall\beta\in P,\;\alpha-\beta\notin P\rbrace$. The goal is to show that $\Delta$ spans $P$ with positive integer coefficients, hence $\Delta$ spans the whole system with integer coefficients, either all positive or negative. $\Delta$ spans the whole vector space, you have to prove that $\Delta$ is also a linearly-independant family of vectors.
All this proof is detailed in the first link (being chapter III-2), and some geometrical descriptions made in III-1 are used.

Afterwards, still in III-2, the author manages to describe all the possible bases as such bases generated using a specific regular vector. There you'll get your positive vector set.
