Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$
n/8<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space spanned by $3\times 4$ matrix
$$N=\begin{bmatrix}
-D&C&0&0\\
-B&0&A&0\\
0&0&-D&C
\end{bmatrix}.$$

Denote the $\Bbb Q$-linear space spanned by rows of $N$ by $T_{A,B,C,D}\subseteq\Bbb Q^4$ and denote the set of non-zero $\Bbb Z$ vectors in $T_{A,B,C,D}$ by $T_{A,B,C,D}[\Bbb Z]^\star$.

>1. Is there a name for the quantity $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ where $\|v\|_\infty$ is largest coordinate by magnitude of vector $v$?

>2. How is $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ distributed as a function of $A,B,C,D$ chosen with the constraints above (at least consider $A,B$ and $C,D$ each a coprime pair and $n<A,B,C,D<2n$) and what is its average value?

Simulations and heuristics suggest a value between $\Omega(n^{1/2})$ and $\Omega(n^{2/3})$ with $\Omega(n^{2/3})$ being the most likely possibility of lower bound for expected value.