Lazard correspondence establishes a 1-isomorphism between Lazard Lie group and Lazard Lie ring
Suppose is a Lazard Lie group and is its Lazard Lie ring with the logarithm map and the exponential map. (These are both bijections, and are inverses of each other).
Then, and are 1-isomorphisms, i.e., they are isomorphisms when restricted to cyclic subgroups.
This follows directly from fact (1), and the fact that the logarithm map is bijective.