Is there an $SO(n,n)$ transformation that makes the Euclidean norm-squared of all the null vectors of the $(n,n)$ hypercubic lattice strictly greater than 4?
Example: Let $\Lambda_{\rm sL}$ denote the (23-dimensional) shorter Leech lattice. The minimal norm-squared of any vector is 3. Therefore, all null vectors of the lattice $\Lambda_{\rm sL} \oplus \Lambda_{\rm sL}$ have Euclidean norm-squared at least 6. Since all odd integral unimodular lattices are equivalent up to an $SO(n,n)$, then the lattice $\Lambda_{\rm sL} \oplus \Lambda_{\rm sL}$ is related to the 46-dimensional hypercubic lattice.
Is there an example with $n < 23$?
