Derivation-invariance is transitive

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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)
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 transitive Lie subring property
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
View other analogues of characteristicity is transitive|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring.

Related facts

Proof

Given: A Lie ring L with Lie subrings A \le B \le L. B is a derivation-invariant Lie subring of L and A is a derivation-invariant subring of B.

To prove: A is a derivation-invariant subring of L.

Proof: Suppose d is a derivation of L.

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

Since A is derivation-invariant in B, d' restricts to a map from A to itself. Thus, d restricts to a map from A to A.