Root systems and root lattices in number fields

The following construction of a root system of type $G_2$ is known (see J.-P. Serre’s book “Lie Algebras and Lie Groups”, page 315 of the Russian translation): “It can be described as the set of algebraic integers of a cyclotomic field generated by a cubic root of unity, with norm 1 and 3”. The talk, based on a joint work with Yu. G. Zarhin, concerns the problem of realization of root systems, their Weyl groups and their root lattices in the form of groups and lattices naturally associated with number fields.