Let $\mathfrak{g}$ be a complex Lie algebra. Then, we have two canonical representations of $\mathfrak{g}$, namely, the adjoint representation on $\mathfrak{g}$ and the coadjoint representation on $\mathfrak{g}^*$.
Question. Are $\mathfrak{g}$ and $\mathfrak{g}^*$ isomorphic as $\mathfrak{g}$-representations?
We know that the answer is 'yes' if $\mathfrak{g}$ is semisimple, since we can use the Killing form to produce an isomorphism.
But this is also true for some non-semisimple Lie algebras. For instance if $\mathfrak{g}=\mathbb{C}\langle x,y\rangle$ with $[x,y]=x$, then the unique skew-symmetric bilinear form $\omega$ with $\omega(x,y)=1$ is $\mathrm{ad}$-invariant and hence produces an isomorphism $\mathfrak{g}\to\mathfrak{g}^*$.
Hence, we could also ask:
Question. Are there conditions on $\mathfrak{g}$ which imply that $\mathfrak{g}$ and $\mathfrak{g}^*$ are isomorphic as $\mathfrak{g}$-representations, and which include both semisimple Lie algebras and the example $\mathbb{C}\langle x,y\rangle$ above?