Lie ring acts as derivations by adjoint action

Statement

Let $\text{[math]}$ be a Lie ring. For any $\text{[math]}$ , define the map:

$\text{[math]}$

given by:

$\text{[math]}$

(this is termed the left adjoint action by $\text{[math]}$ ).

Then, the following are true:

For every $\text{[math]}$ , $\text{[math]}$ is a derivation of $\text{[math]}$ (a derivation arising this way is termed an inner derivation).
The map from $\text{[math]}$ to the Lie ring of derivations of $\text{[math]}$ , that sends an element $\text{[math]}$ to the derivation $\text{[math]}$ , is a homomorphism of Lie rings.