Invariance under any derivation with partial divided Leibniz condition powers is transitive

Statement
Suppose $$L$$ is a Lie ring and $$A,B$$ are Lie subrings of $$L$$ with $$A$$ contained in $$B$$. Suppose $$B$$ is a Lie subring invariant under any derivation with partial divided Leibniz condition powers in $$L$$. Similarly, suppose $$A$$ is a Lie subring invariant under any derivation with partial divided Leibniz condition powers in $$B$$.

Then, $$A$$ is a Lie subring invariant under any derivation with partial divided Leibniz condition powers in $$L$$.

Related facts

 * Derivation-invariance is transitive