Lie subring whose sum with any subring is a subring

Definition
Suppose $$L$$ is a Lie ring and $$S$$ is a subring of $$L$$. We say that $$S$$ is a Lie subring whose sum with any subring is a subring if, for any Lie subring $$A$$ of $$L$$, the subgroup $$S + A$$ is also a Lie subring of $$L$$.

Stronger properties

 * Weaker than::Ideal of a Lie ring:
 * Weaker than::Subring of a Lie ring that is maximal as a subgroup

Metaproperties
A join of Lie subrings with this property also has this property.