Fully invariant subgroup of additive group of a Lie ring

Definition
Let $$L$$ be a Lie ring. A subset $$S$$ of $$L$$ is termed a fully invariant subgroup of the additive group of $$L$$ if it satisfies the following equivalent conditions:


 * 1) Consider the additive group of $$L$$. Then, $$S$$ is a defining ingredient::fully invariant subgroup of the additive group of $$L$$.
 * 2) $$S$$ is a Lie subring of $$L$$ that is also fully invariant as an additive subgroup of $$L$$.