Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \mathrm{Der}(\frak{a})$. For the zero map $\frak{g} \to \mathrm{Der}(\frak{a})$ we get the usual direct product. But can there exist more than one non-zero map $\frak{g} \to \frak{a}$, giving two non-isomorphic semi-direct product? What is a motivating example? I will be interested most in a example in the context of Levi's theorem.



Non-isomorphic direct products of a solvable and a semisimple Lie algebra
Let $K$ be the ground field of characteristic zero.


For $n\ge 1$, let $\mathfrak{v}\_n$ denote an $n$-dimensional irreducible representation of $\mathfrak{sl}\_2(K)$ over $K$ ($(n-1)$-th symmetric power of the standard 2-dimensional representation). Then $\mathfrak{v}\_3\rtimes\mathfrak{sl}\_2(K)$ and ($\mathfrak{v}\_1\oplus\mathfrak{v}\_2)\rtimes\mathfrak{sl}\_2(K)$ are non-direct semidirect products and are not isomorphic (the first is perfect and not the second).


Note: however, in general (say over the complex or real numbers), there are only finitely many homomorphisms from $\mathfrak{g}$ to $\mathrm{Der}(\mathfrak{a})$ up to conjugacy, and hence there are only finitely many isomorphism classes of semidirect products.


