Lie subring of index two not is ideal
Opposite facts for groups
Opposite facts for Lie rings
Consider a Klein four-group, i.e., an elementary abelian group of order four. This has four elements, . Define the Lie bracket as . This forces all Lie brackets of distinct nonzero elements to equal and all Lie brackets involving an element and itself or zero to be zero. This is clearly a Lie ring.
Consider the subring . This is clearly a Lie ring, since by definition. It has index two as a subgroup. However, it is not an ideal since .