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 $L$.