Example 1

Let $g$ be a Lie algebra over $K$ and $D$ a derivation of $g$. Let $h$ be the commutative Lie algebra $K$. The mapping $\lambda \mapsto \lambda D (\lambda \in K)$ is a homomorphism of $h$ into the Lie algebra of derivations of $g$. We form the corresponding semi-direct product $t$ of $h$ by $g$. Let $x_0$ be the element $(0, 1)$ of $t$. For all $x \in g$, $Dx = [x_0, x]$.