I begin with a definition of positive root systems of a root system over Euclidean space.
A subset $\Delta$ of root system $\Phi$ is called a simple root system (or base) in $\Phi$ if
(1) $\Delta$ is a basis for $V$
(2) For any root $\beta \in \Phi$ $$ \beta = \sum_{\alpha \in \Delta} m_\alpha \alpha $$ with all $m_\alpha \in \Bbb Z_{\geq 0}$ or all $m_\alpha \in \Bbb Z_{\leq 0}$.
The root $ \beta = \sum_{\alpha \in \Delta} m_\alpha \alpha $ is positive if $m_\alpha \in \Bbb Z_{\geq 0}$, and negative if $m_\alpha \in \Bbb Z_{\leq 0}$. The set of all positive roots (the positive root system) associated to $\Delta$ will be denoted by $\Phi^+$.
I want to prove the following:
Assume that $\Phi'$ is a subset of $\Phi$. Prove that $\Phi'$ is the positive root system of $\Phi$ if and only if it satisfies the following properties:
(1) For each root $\alpha\in\Phi$ exactly one of the roots $\alpha, -\alpha$ is contained in $\Phi'.$
(2) For any two distinct $\alpha, \beta\in \Phi'$ such that $\alpha+\beta$ is a root, $\alpha+\beta\in\Phi'.$
Proof of the necessary condition is straightforward. Any help for the sufficient condition would be appreciated. Thank you.
I had found an argument, but actually your question is answered in Bourbaki, Groupes et algèbres de Lie, Ch. VI, 1.7, Cor. 1.
My argument was as follows. One definition of positive root system is: given a hyperplane which does not contain any root, take the intersection of the root system with either half-space delimited by the hyperplane.
If we can show that $\Phi'$ is contained in an open half-space whose boundary contains no roots, then we are done, because then $\Phi'$ is contained is the positive root system defined by that half-space, and both sets have the same cardinality, by condition (2).
Now for the half-space, choose the points which have a positive pairing with $$ \rho^\vee := \frac 1 2 \sum_{\beta\in\Phi'} \beta^\vee. $$ So we have to show that $\langle \alpha, \rho^\vee \rangle > 0$ for all $\alpha\in\Phi'$ (actually this pairing will be equal to the height of $\alpha$ for this choice of positive roots). By condition (2), the pairing will be negative for the other roots, so no root will lie on the hyperplane defined by $\rho^\vee$. We have $$ \langle \alpha, \rho^\vee \rangle = 1 + \frac 1 2 \sum_{\beta\in\Phi'-\{\alpha\}} \langle \alpha, \beta^\vee \rangle. $$ The roots non-proportional to $\alpha$ are partitioned into $\alpha$-strings: see Bourbaki Ch. VI, 1.3, Prop. 9. Such a string is symmetric with respect to the hyperplan $\alpha^\vee = 0$. By condition (1), the intersection of the string with $\Phi'$ is some possibily smaller interval with the same end. Combining this information, one sees that each string intersected with $\Phi'$ contributes non-negatively to the sum. This finishes the proof.
I hope it is clear enough. It would be easier to explain with a picture!