Proposition 1

*Let g be a Lie algebra. For all $x \in g$, $\operatorname{ad} x$ is a derivation. The mapping $x \mapsto \operatorname{ad} x$ is a homomorphism of the Lie algebra g into the Lie algebra $\mathfrak{d}$ of derivations of g. If $D \in \mathfrak{d}$ and $x \in g$, $[D, \operatorname{ad} x] = \operatorname{ad}(Dx)$.

Identity (4) can be written:

or:

whence the first two assertions. On the other hand, if $D \in \mathfrak{d}$, $x \in g$, $y \in g$, then $[D, \operatorname{ad} x] \cdot y = D([x, y]) - [x, Dy] = [Dx, y] = (\operatorname{ad} Dx) \cdot y$, whence the last assertion.

The mapping $\operatorname{ad} x$ is also called the inner derivation defined by $x$.