Derivation-invariance is transitive

Statement
A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring. In symbols, if $$L$$ is a Lie ring with subrings $$A,B$$ such that $$B$$is a derivation-invariant Lie subring of $$L$$ and $$A$$ is a derivation-invariant Lie subring of $$B$$, then $$A$$ is a derivation-invariant Lie subring of $$L$$.

Related facts

 * Derivation-invariant subring of ideal implies ideal
 * Invariance under any derivation with partial divided Leibniz condition powers is transitive

Proof
Given: A Lie ring $$L$$ with Lie subrings $$A \le B \le L$$. $$B$$ is a derivation-invariant Lie subring of $$L$$ and $$A$$ is a derivation-invariant subring of $$B$$.

To prove: $$A$$ is a derivation-invariant subring of $$L$$.

Proof: Suppose $$d$$ is a derivation of $$L$$.

Since $$B$$ is a derivation-invariant subring of $$L$$, $$d$$ restricts to a map from $$B$$ to itself. Let $$d'$$ be the restriction of $$d$$ to $$B$$. Clearly, $$d'$$ is a derivation of $$B$$.

Since $$A$$ is derivation-invariant in $$B$$, $$d'$$ restricts to a map from $$A$$ to itself. Thus, $$d$$ restricts to a map from $$A$$ to $$A$$.