Lie subring of index two not is ideal

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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.

Related facts

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, 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.