Derivation-invariance is Lie bracket-closed

Statement
Suppose $$L$$ is a Lie ring and $$A,B$$ are two derivation-invariant Lie subrings of $$L$$. Then, the subring:

$$[A,B] := \langle [a,b] \mid a \in A, b \in B \rangle$$

is a derivation-invariant subring. In other words, if $$d:L \to L$$ is a derivation, then $$d[A,B] \le [A,B]$$.

Analogues

 * Characteristicity is commutator-closed: The commutator of two characteristic subgroups of a group is again characteristic.