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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Statement

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

Related facts

• Derivation-invariant subring of ideal implies ideal

Proof

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

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 $d^{\prime }$ be the restriction of $d$ to $B$. Clearly, $d^{\prime }$ is a derivation of $B$.

Since $A$ is derivation-invariant in $B$, $d^{\prime }$ restricts to a map from $A$ to itself. Thus, $d$ restricts to a map from $A$ to $A$.