Lie subring invariant under any additive endomorphism satisfying a comultiplication condition

Statement
Suppose $$L$$ is a Lie ring and $$S$$ is a Lie subring of $$L$$. We say that $$S$$ is invariant under any additive endomorphism satisfying a comultiplication condition if, for any $$f:L \to L$$ that is an additive endomorphism satisfying a comultiplication condition, we have $$f(S) \subseteq S$$.