Fully invariant Lie subring

Definition
A subring $$S$$ of a Lie ring $$L$$ is termed a fully invariant Lie subring if, for every endomorphism $$\sigma$$ of $$L$$, $$\sigma(S) \subseteq S$$.

Lazard Lie ring
Suppose $$G$$ is a Lazard Lie group and $$L$$ is a Lazard Lie ring. Under the natural bijection between $$L$$ and $$G$$, fully invariant subrings of $$L$$ correspond to fully invariant subgroups of $$G$$.