# Lie subring of index two not is ideal

From Groupprops

## Contents

## Statement

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

## Related facts

### Opposite facts for groups

### Opposite facts for Lie rings

## Proof

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 .