Ideal property is centralizer-closed
This article gives the statement, and possibly proof, of a Lie subring property (i.e., ideal of a Lie ring) satisfying a Lie subring metaproperty (i.e., centralizer-closed Lie subring property)
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Analogue for group theory
The general version of this result is: invariance under any set of derivations is centralizer-closed, which reduces to this result when we take the inner derivations (see Lie ring acts as derivations by adjoint action). Another special case of this general result is:
Proof using the notion of inner derivations
Hands-on proof using Jacobi identity
Given: A Lie ring , an ideal of . is the centralizer of in .
To prove: is an ideal of , i.e., for any and , .
Proof: To show it suffices to show that for all .
By the Jacobi identity:
Since centralizes , , so the second term is zero. Further, since is an ideal, , and since centralizes , . Thus, both the second and third term on the left side are zero, forcing .