Characteristicity is transitive for Lie rings

This article gives the statement, and possibly proof, of a Lie subring property (i.e., characteristic subring of a Lie ring) 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 characteristic subring of a Lie ring |Get facts that use property satisfaction of characteristic subring of a Lie ring | Get facts that use property satisfaction of characteristic subring of a Lie ring|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: derivation-invariance is transitive
View other analogues of characteristicity is transitive|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

If $A\le B\le L$ are Lie rings such that $A$ is a characteristic subring of $B$ and $B$ is a characteristic subring of $L$, then $A$ is a characteristic subring of $L$.

Related facts

• Derivation-invariance is transitive
• Derivation-invariant subring of ideal implies ideal