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 $L$.

Related facts

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 $[x,y]=x$.