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.