Fully invariant derivation-invariant Lie subring

Definition
Suppose $$L$$ is a Lie ring and $$S$$ is a subring of $$L$$. We say that $$S$$ is a fully invariant derivation-invariant Lie subring of $$L$$ if $$S$$ is a defining ingredient::fully invariant Lie subring and is also a defining ingredient::derivation-invariant Lie subring of $$L$$.