Corollary 6

We adopt the hypotheses and notation of Prop. 20. Let $V_1$ be the vector subspace of V generated by $\Sigma$. Then $P \cap (-P) = Q \cup (-Q) = V_1 \cap R$ is a root system in $V_1$ with basis $\Sigma$.

We have $P \cap (-P) = (R_+(C) \cup Q) \cap ((-R_+(C)) \cup (-Q)) = Q \cup (-Q)$. Th. 3 proves that $Q \cup (-Q) = V_1 \cap R$. Finally, $\Sigma$ is a basis of the root system $V_1 \cap R$ by Cor. 4.