Does there exist $\xi_1,\ldots,\xi_4\in\mathbb R$ such that
$$\forall (\alpha,\beta)\in\mathbb Q^2\setminus\{0,0\},\quad \begin{cases}\dim_{\mathbb Q}(\alpha \xi_1+\beta\xi_3,\,\alpha\xi_2+\beta\xi_4)= 2 \\ \dim_{\mathbb Q}(\xi_1,\xi_2,\xi_3,\xi_4,\xi_1\xi_4-\xi_2\xi_3)\leqslant 2\quad ?\end{cases}$$
I believe the answer to be negative, but I have no idea in how to prove such a result isn't true.
I have try to search many examples (while I still have hoped for it to be true), but non of those seem to work for various reasons. For instance, I searched quite a lot something like: $\xi_i=a_i\xi+b_i\xi^2$ for $i\in\{1,\ldots,4\}$, but it was unsuccessful.
Any progress regarding this problem would be much appreciated.
Without some restrictions the answer in the upper case is clearly negative: just take $\;\alpha=\beta=0\;$, and thus the overall answer is no .