Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant
- 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.
- Homomorph-containing subgroup of additive group of Lie ring is self-derivation-invariant and homomorph-containing
- Characteristic subgroup of additive group of odd-order Lie ring is derivation-invariant and fully invariant
(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.