Ideal-large Lie subring

Definition
Let $$L$$ be a Lie ring. A subring $$S$$ of $$L$$ is termed an ideal-large Lie subring if, for any nonzero ideal $$I$$ of $$L$$, $$S \cap I$$ is also an ideal of $$L$$.

Facts

 * Nilpotent implies center is ideal-large