Solution
= Solution
The <Lie bracket> is bilinear, so $\operatorname{ad}x$ is a <linear map>. Rearranging the <Jacobi identity> gives
$$
[x,[y,z]]=[[x,y],z]+[y,[x,z]].
$$
Thus
$$
\boxed{(\operatorname{ad}x)[y,z]=[(\operatorname{ad}x)y,z]+[y,(\operatorname{ad}x)z],}
$$
which is the defining Leibniz rule for a <derivation of a Lie algebra>. In particular $\operatorname{ad}x$ is an <inner derivation of a Lie algebra>.