Are there finite-dimensional Lie algebras which are not defined over the integers?

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and let $R \subset \mathbb{C}$ be a subring. Say that $\mathfrak{g}$ is defined over $R$ if there exists a basis $x_1, ... x_n$ for $\mathfrak{g}$ such that the structure constants $c_{ijk}$ of the bracket $$[x_i, x_j] = \sum_k c_{ijk} x_k$$ all lie in $R$. It is classical that all semisimple $\mathfrak{g}$ are defined over $\mathbb{Z}$. But this is also true for some non-semisimple $\mathfrak{g}$ such as the Lie algebra of $n \times n$ upper triangular or strictly upper triangular matrices.

In fact, I don't know an example of such a $\mathfrak{g}$ which isn't defined over $\mathbb{Z}$ although I would be surprised if they didn't exist. Can someone construct one or prove that they don't exist? If they do exist, is a weaker statement true? For example, are all such $\mathfrak{g}$ defined over a number field?

Three dimensional Lie algebras (over $\mathbb{C}$, say) vary in moduli: see for instance this paper for a description. (I am not conversant with the details here...) In particular, one of the connected components of the moduli space has dimension one, so the generic point of this moduli space cannot be defined over any algebraic extension of $\mathbb{Q}$.

Ah, there it is. In one of Pete Clark's answers he links to a paper giving a continuous family $L_a^3$ of pairwise non-isomorphic solvable $3$-dimensional Lie algebras over any field. Explicitly, these are spanned by $x_1, x_2, x_3$ satisfying $$[x_3, x_1] = x_2, [x_3, x_2] = ax_1 + x_2$$

for a parameter $a$ (and, I suppose, $[x_1, x_2] = 0$). Since $\mathbb{C}$ is uncountable the conclusion follows.