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Groupprops β

Derivation-invariance is transitive

Statement

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

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.