One can classify forms over an arbitrary subfield $K$ of $\mathbf{C}$. These are, beyond the obvious one ($K\ltimes K^2$) "split form", the $L^*\ltimes L$ where $L$ ranges over quadratic extensions of $K$. In particular, the only nonsplit real form is the Lie algebra of $\mathbf{C}^*\ltimes\mathbf{C}$, viewed as 4-dimensional real Lie algebra.

The proof: let $\mathbf{g}$ be a 4-dimensional Lie algebra over $K$ which has the given complexification. So $\mathbf{g}$ has 2-dimensional and abelian derived subalgebra $\mathfrak{a}$, which is the constant term in the lower central series. Let $\mathbf{h}$ be a Cartan subalgebra (= nilpotent self-normalizing) subalgebra. It is known (see e.g. Bourbaki) that $\mathfrak{g}=\mathfrak{h}+\mathfrak{a}$. Since at the complex level, there is nilpotent subalgebra of dimension $\ge 3$, we deduce that $\mathfrak{h}$ is 2-dimensional, and hence $\mathfrak{g}=\mathfrak{h}\ltimes\mathfrak{a}$. The action of $\mathfrak{h}$ on $\mathfrak{a}$ is faithful (as we see at the complex level). So to classify these algebras, we have to classify 2-dimensional abelian subalgebras of $\mathfrak{gl}_2(K)$ up to conjugation. Moreover it is enough to consider those acting semisimply (to get the right complexification). Then we get: subfields (irreducible case), diagonal matrices (reducible case). This corresponds to the given description. Conversely when complexifying, one can diagonalize the action and we see that we get the Lie algebra of $(\mathbf{C}^*\ltimes\mathbf{C})^2$.