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?