# 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)
View all Lie subring metaproperty satisfactions | View all Lie subring metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for Lie subring properties
Get more facts about derivation-invariant Lie subring |Get facts that use property satisfaction of derivation-invariant Lie subring | Get facts that use property satisfaction of derivation-invariant Lie subring|Get more facts about Lie bracket-closed Lie subring property
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 $d[A,B] \le [A,B]$.