Cartan subgroups and regular points of o‐minimal groups

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G.When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.