Root systems in number fields

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal{L}(K)$ generated by $\mathrm{A}\mathrm{u}\mathrm{t}(K)$ and multiplications by the elements of $K^{\ast }$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\mathcal{L}(K)$ for a number field $K$ of degree $n$ over $\mathbb{Q}$.