Lie subring of index two not is ideal

Statement
There can exist a Lie ring $$L$$ and a subring $$S$$ of $$L$$ such that $$S$$ has index two in $$L$$, but $$S$$ is not an ideal in $$L$$.

Opposite facts for groups

 * Subgroup of index two is normal

Opposite facts for Lie rings

 * Nilpotent implies every maximal subring is an ideal

Proof
Consider a Klein four-group, i.e., an elementary abelian group of order four. This has four elements, $$0,x,y,x+y$$. Define the Lie bracket as $$[x,y] = x$$. This forces all Lie brackets of distinct nonzero elements to equal $$x$$ and all Lie brackets involving an element and itself or zero to be zero. This is clearly a Lie ring.

Consider the subring $$\{ 0, y \}$$. This is clearly a Lie ring, since $$[y,y] = 0$$ by definition. It has index two as a subgroup. However, it is not an ideal since $$[x,y] = x$$.