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

This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., Lie subring whose sum with any subring is a subring) need not satisfy the second Lie subring property (i.e., ideal of a Lie ring)
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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. 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 $S$.