Derivation-invariant subring of ideal implies ideal

Statement
Suppose $$L$$ is a Lie ring, $$A$$ is an ideal of $$L$$, and $$B$$ is a derivation-invariant subring of $$A$$. Then, $$B$$ is also an ideal of $$L$$.

Analogues

 * Characteristic of normal implies normal: A characteristic subgroup of a normal subgroup is normal. Here, characteristic subgroup plays the role analogous to derivation-invariant subring, and normal subgroup plays a role analogous to ideal.