Derivation-invariance is Lie bracket-closed

This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant Lie subring) satisfying a Lie subring metaproperty (i.e., Lie bracket-closed Lie subring property)
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ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity is commutator-closed.
Another analogue to the same fact, in the same new context, is: characteristicity is Lie bracket-closed
View other analogues of characteristicity is commutator-closed|View other analogues from group to Lie ring (OR, View as a tabulated list)

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 $d[A,B]\leq [A,B]$.

Related facts

Analogues

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