Subring of a Lie ring

Definition
Let $$L$$ be a Lie ring and $$S$$ be a subset of $$L$$. We say that $$S$$ is a subring of $$L$$, or a Lie subring, if it satisfies the following equivalent conditions:


 * 1) $$S$$ is a subgroup of the additive group of $$L$$ and is closed under the Lie bracket of $$L$$.
 * 2) $$S$$ is a Lie ring with the addition and Lie bracket operations induced from $$L$$.