Lazard correspondence establishes a correspondence between abelian subgroups and abelian subrings

Statement
Suppose $$G$$ is a Lazard Lie group, $$L$$ is its Lazard Lie ring, and $$\exp:L \to G$$ and $$\log:G \to L$$ are the exponential and logarithm maps respectively (they are both bijections and are inverses of each other). Note that we may wish to think of $$G$$ and $$L$$ as having the same underlying set and treat the bijections as being the identity map on the underlying set; however, for conceptual convenience, we are using separate symbols for the group and Lie ring and explicit names for the bijections.

This bijection establishes a correspondence:

abelian subgroups of $$G$$ $$\leftrightarrow$$ abelian subrings of $$L$$

Related facts

 * Lazard correspondence establishes a correspondence between Lazard Lie subgroups and Lazard Lie subrings
 * Lazard correspondence establishes a correspondence between abelian normal subgroups and abelian ideals
 * Lazard correspondence establishes a correspondence between powering-invariant subgroups and powering-invariant subrings