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.