# Derivation-invariance is transitive

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Revision as of 22:27, 8 October 2008 by Vipul (talk | contribs) (New page: ==Statement== A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring. ==Related facts== * [[Derivation-invariant subring of i...)

## 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 with Lie subrings . is a derivation-invariant Lie 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 .