Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:

1. For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \geq 0$ such that $c_1 \lambda_1 + c_2 \lambda_2 \in \Phi$, we have $c_1 \lambda_1 + c_2 \lambda_2 \in \Phi'$.

2. For all $\lambda \in \Phi'$, we have $-\lambda \notin \Phi'$.

One example would be a choice of positive roots.

Is there a term for such subsets? I'm not an expert in root systems or Lie theory, so I might be missing something obvious.