# Derivation-invariance is Lie bracket-closed

From Groupprops

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 is a Lie ring and are two derivation-invariant Lie subrings of . Then, the subring:

is a derivation-invariant subring. In other words, if is a derivation, then .

## Related facts

### Analogues

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