Lazard correspondence establishes a correspondence between Lazard Lie subgroups and Lazard Lie 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:

Lazard Lie subgroups of $$G$$ $$\leftrightarrow$$ Lazard Lie subrings of $$L$$

Facts used

 * 1) uses::Lazard correspondence defines isomorphism of categories over Set between full subcategories of the category of groups and the category of Lie rings

Proof
The proof follows from Fact (1).

Refinement of correspondence by subgroup and subring properties
The correspondence can be refined to a correspondence between certain kinds of subgroups (defined as those satisfying a particular subgroup property) and certain kinds of subrings (defined as those satisfying a particular Lie subring property). Some examples are described below.