# External semidirect product of Lie rings

From Groupprops

## Definition

### Definition with the left action convention

Suppose is a Lie ring and is a Lie ring along with a homomorphism of Lie rings from to the Lie ring of derivations . The **external semidirect product of Lie rings**, denoted , is defined as follows:

- As an abelian group, it is the external direct product of and , denoted or .
- The Lie bracket is as follows:

where denotes the image of under the derivation .

### Why the action convention does not matter

This is basically because every Lie ring is naturally isomorphic to its opposite Lie ring via the negative map.