Subring of a Lie ring that is maximal as a subgroup

Definition
Let $$L$$ be a Lie ring and $$S$$ be a subring of $$L$$. We say that $$S$$ is maximal as a subgroup if the additive group of $$S$$ is a maximal subgroup of the additive group of $$L$$.

Weaker properties

 * Stronger than::Lie subring whose sum with any subring is a subring
 * Stronger than::Maximal Lie subring