# Derivation-invariance is transitive

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., transitive 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 transitive.

Another analogue to the same fact, in the same new context, is: characteristicity is transitive for Lie rings

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## Statement

A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring. In symbols, if is a Lie ring with subrings such that is a derivation-invariant Lie subring of and is a derivation-invariant Lie subring of , then is a derivation-invariant Lie subring of .

## Related facts

## Proof

**Given**: A Lie ring with Lie subrings . is a derivation-invariant Lie subring of and is a derivation-invariant subring of .

**To prove**: is a derivation-invariant subring of .

**Proof**: Suppose is a derivation of .

Since is a derivation-invariant subring of , restricts to a map from to itself. Let be the restriction of to . Clearly, is a *derivation* of .

Since is derivation-invariant in , restricts to a map from to itself. Thus, restricts to a map from to .