$\text{SL}(n,\mathbb Z)$ acts transitively on the set of ordered pairs of distinct 1-dimensional subspaces of $\mathbb Q^n$.
Could you mention an article or a book where such a proof can be found? Would you sketch such a proof here? Thanks in advance for any help.
This is false. For instance, there is no element of $\text{SL}(2,\mathbb{Z})$ taking the pair of 1d subspaces $\{\mathbb{Q} \cdot (1,0),\mathbb{Q} \cdot (0,1)\}$ to the pair $\{\mathbb{Q} \cdot (2,1),\mathbb{Q} \cdot (0,1)\}$. Similar counterexamples exist in higher dimensions.
