Homomorph-containing subgroup of additive group of Lie ring is self-derivation-invariant and homomorph-containing

Statement
Suppose $$L$$ is a Lie ring, and $$A$$ is a subgroup of the additive group of $$L$$ such that $$A$$ is a homomorph-containing subgroup of $$L$$. Then, the following three conditions hold:


 * 1) $$A$$ is a Lie subring of $$L$$.
 * 2) $$A$$ is a self-derivation-invariant Lie subring of $$L$$.
 * 3) $$A$$ is a homomorph-containing Lie subring of $$L$$.

Related facts

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