Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras and let $\theta$ be a homomorphism of Lie algebras from $\mathfrak{g}$ to $\operatorname{Der}(\mathfrak{h})$. We can then form the semidirect product $\mathfrak{g} \ltimes_\theta \mathfrak{h}$. Its underlying vector space is the direct sum $\mathfrak{g} \oplus \mathfrak{h}$, and its Lie bracket is $$ [(x_1, y_1), (x_2, y_2)] = ( [x_1, x_2], \theta(x_1)(y_2) - \theta(x_2)(y_1) + [y_1, y_2] ) \,. $$
Can we describe the Lie algebra of derivations $\operatorname{Der}( \mathfrak{g} \ltimes_\theta \mathfrak{h} )$ is a sensible way? If so, can the necessary computations be done is a “clever” way?
As as special case of the above problem I have calculated $\operatorname{Der}(\mathfrak{g} \oplus \mathfrak{h})$. For this I’ve taken an arbitrary vector space endomorphism $\delta$ of $\mathfrak{g} \oplus \mathfrak{h}$ and written it in matrix form $$ \delta \equiv \begin{pmatrix} \delta_{11} & \delta_{12} \\ \delta_{21} & \delta_{22} \end{pmatrix} \,, $$ so that $$ \delta( (x,y) ) = ( \delta_{11}(x) + \delta_{12}(y), \delta_{21}(x) + \delta_{22}(y) ) \,. $$ That $\delta$ is a dervation of $\mathfrak{g} \oplus \mathfrak{h}$ can then be equivalently expressed by conditions on the matrix entries $\delta_{ij}$, which then gives a sensible description of $\operatorname{Der}(\mathfrak{g} \oplus \mathfrak{h})$. (One finds that $\delta_{11}$ and $\delta_{22}$ must be derivations, and that $\delta_{12}$ and $\delta_{21}$ need to be homomorphism of Lie algebras from one direct summand into the center of the other direct summand.)
I understand that I can apply the same approach for the semidirect product $\mathfrak{g} \ltimes_\theta \mathfrak{h}$, and that I will end up with some equivalent conditions on the matrix entries $\delta_{ij}$. However, the calculations for the direct sum (i.e. the case $\theta = 0$) were already kinda annoying, so I’m hoping that there is a better way. It is also not clear that the resoluting conditions on the entries $\delta_{ij}$ for $\operatorname{Der}(\mathfrak{g} \ltimes_\theta \mathfrak{h})$ will be something that I can interpret as something sensible.