Let $\mathfrak{g}$ be a Lie algebra. If $\mathfrak{h}$ is another Lie algebra and $\theta$ is a homomorphism of Lie algebras from $\mathfrak{h}$ to $\operatorname{Der}(\mathfrak{g})$, then we can form the semidirect product $\mathfrak{h} \ltimes_\theta \mathfrak{g}$. As a special case of this construction, one can choose $\mathfrak{h} = \mathfrak{g}$ and $\theta = \mathrm{ad}$.
How is the Lie algebra $\mathfrak{g} \ltimes_{\mathrm{ad}} \mathfrak{g}$ called?
The same question could be asked about groups, but I also don’t know how the analogous construction for groups is called.