Fully invariant implies ideal for class two Lie ring

Statement
Suppose $$L$$ is a fact about::Lie ring of nilpotency class two and $$A$$ is a fact about::fully invariant Lie subring of $$L$$. Then, $$A$$ is an ideal of $$L$$.

Related facts

 * Characteristic not implies ideal
 * Characteristic not implies derivation-invariant
 * Inner derivation implies endomorphism for class two Lie ring
 * Derivation equals endomorphism for Lie ring iff it is abelian

Facts used

 * 1) uses::Inner derivation implies endomorphism for class two Lie ring

Proof idea
By fact (1), invariance under all endomorphisms implies invariance under inner derivations, so fully invariant subrings are ideals.