Lie subring whose sum with any subring is a subring not implies ideal

Statement
There can exist a Lie ring $$L$$ and a subring $$S$$, such that the sum of $$S$$ and any subring of $$L$$ is a subring, but $$S$$ is not an ideal.

Facts used

 * 1) uses::Subring of a Lie ring that is maximal as a subgroup

Proof
Let $$L$$ be the Klein four-group with $$x,y$$ two of the non-identity elements. The four elements of $$L$$ are $$0,x,y,x+y$$. We define the Lie bracket as follows:

$$[x,y] = x$$

This completely determines the Lie bracket:

$$[x,x+y] = [y,x] = [x+y,x] = [y,x+y] = [x+y,y] = x$$.

Let $$S$$ be the subring of $$L$$ defined as $$\{ 0,y \}$$. The sum of $$S$$ with any subring of $$L$$ that does not contain $$S$$ is the whole ring $$L$$ (since $$S$$ is maximal as a subgroup of $$L$$). Also, the sum of $$S$$ with any subring of $$L$$ contained in $$S$$ is $$S$$. Thus, in any case, the sum of $$S$$ with any subring is a subring. (for more, see fact (1)).

However, $$S$$ is not an ideal of $$L$$, because $$[y,x] = x$$, and $$x$$ is not in $$S$$.