# Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant

## Statement

Suppose is a Lie ring. Suppose is a Fully invariant subgroup (?) of the additive group of . Then:

- is a Lie subring of , i.e., is closed under the Lie ring operations of .
- is a Derivation-invariant Lie subring (?) of , i.e., any derivation of sends to itself.
- is a Fully invariant Lie subring (?) of : any endomorphism of as a Lie ring sends to itself.

## Related facts

### Similar facts

### Applications

## Proof

(Using notation as in the statement above).

Since is a subgroup of by definition, proving (1) and (2) only requires us to show that is closed under arbitrary derivations (this would imply *both* (1) and (2) since the Lie bracket with an element of is itself an inner derivation). This, in turn, follows from the fact that any derivation is an endomorphism of the underlying abelian group structure, and , being fully invariant, is thus sent to itself by this endomorphism.

For (3), note that any endomorphism of as a Lie ring is also an endomorphism of the underlying abelian group structure of . Since is fully invariant, the endomorphism must send to within itself.