Characteristicity is transitive for Lie rings

Statement
If $$A \le B \le L$$ are Lie rings such that $$A$$ is a characteristic subring of $$B$$ and $$B$$ is a characteristic subring of $$L$$, then $$A$$ is a characteristic subring of $$L$$.

Related facts

 * Derivation-invariance is transitive
 * Derivation-invariant subring of ideal implies ideal