Characteristic subgroup of additive group of odd-order Lie ring is derivation-invariant and fully invariant

Statement
Suppose $$L$$ is a Lie ring of odd order and $$S$$ is a fact about::characteristic subgroup of the additive group of $$L$$. Then:


 * 1) $$S$$ is a Lie subring of $$L$$, i.e., $$S$$ is closed under the Lie bracket of $$L$$.
 * 2) $$S$$ is a fact about::derivation-invariant Lie subring of $$L$$. In particular, $$S$$ is an ideal in $$L$$.
 * 3) $$S$$ is a fact about::fully invariant Lie subring of $$L$$, i.e., every endomorphism of $$L$$ as a Lie ring sends $$S$$ to within itself.

Facts used

 * 1) uses::Characteristic equals fully invariant in odd-order abelian group
 * 2) uses::Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant

Proof
The proof follows from facts (1) and (2).