# Fully invariant implies ideal for class two Lie ring

From Groupprops

## Statement

Suppose is a Lie ring of nilpotency class two (?) and is a Fully invariant Lie subring (?) of . Then, is an ideal of .

## 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

## Proof

### Proof idea

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

### Proof details

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