# Lie ring acts as derivations by adjoint action

From Groupprops

## Statement

Let be a Lie ring. For any , define the map:

given by:

(this is termed the *left adjoint action* by ).

Then, the following are true:

- For every , is a derivation of (a derivation arising this way is termed an inner derivation).
- The map from to the Lie ring of derivations of , that sends an element to the derivation , is a homomorphism of Lie rings.