# Inner derivation implies endomorphism for class two Lie ring

From Groupprops

ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: class two implies commutator map is endomorphism.

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## Contents

## Statement

Suppose is a Lie ring of nilpotency class two (?). In other words, for all . Then, any inner derivation of is an endomorphism of as a Lie ring.

## Related facts

### Applications

### Opposite facts

- Derivation equals endomorphism for Lie ring iff it is abelian
- Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant

## Proof

**Given**: A Lie ring of nilpotency class two, an element .

**To prove**: The map is an endomorphism of as a Lie ring.

**Proof**: The map is clearly an endomorphism of the additive group of , so it suffices to show that it preserves the Lie bracket. In other words, we need to show that for , we have:

.

But since has nilpotency class two, both sides are zero, so the result holds.