On the envelopes of Abelian subgroups in connected Lie groups

An Abelian subgroup $A$ in a Lie group $G$ is said to be regular if it belongs to a connected Abelian subgroup $C$ of the group $G$ (then $C$ is called an envelope of $A$). A strict envelope is a minimal element in the set of all envelopes of the subgroup $A$. We prove a series of assertions on the envelopes of Abelian subgroups. It is shown that the centralizer of a subgroup $A$ in $G$ is transitive on connected components of the space of all strict envelopes of $A$. We give an application of this result to the description of reductions of completely integrable equations on a torus to equations with constant coefficients.