  Let $\Sigma$ be a (reduced) root system. Let $\mathsf{k}$ be an algebraically
closed field of zero characteristic, and consider the corresponding semisimple
Lie algebra $\mathfrak{g}_{\mathsf{k}, \Sigma}$. Then there is a first-order
sentence $\phi_\Sigma$ in the language $\mathcal{L}=(1,0,+,*,-)$ of rings such
that, for any algebraically closed field $\mathsf{k}$ of characteristic 0, the
validity of the Gelfand-Kirillov Conjecture for $\mathfrak{g}_{\mathsf{k},
\Sigma}$ is equivalent to $ACF_0 \vdash \phi_\Sigma$.
