# Lie ring acts as derivations by adjoint action

## Statement

Let $L$ be a Lie ring. For any $x \in L$, define the map:

$\operatorname{ad}_x:L \to L$

given by:

$\operatorname{ad}_x(y) := [x,y]$

(this is termed the left adjoint action by $x$).

Then, the following are true:

• For every $x \in L$, $\operatorname{ad}_x$ is a derivation of $L$ (a derivation arising this way is termed an inner derivation).
• The map from $L$ to the Lie ring of derivations of $L$, that sends an element $x$ to the derivation $\operatorname{ad}_x$, is a homomorphism of Lie rings.