RGD-systems over $\mathbb{F}_2$

In this paper we prove that an RGD-system over $\mathbb{F}_2$ with prescribed commutation relations exists if and only if the commutation relations are Weyl-invariant and can be realized in the group $U_+$. This result gives us a machinery to produce new examples of RGD-systems with complicated commutation relations. We also discuss some applications of this result.