Generation of isometries of certain \(\mathbb{Z}\)-lattices by symmetries (Q1573779)

Positive \(\mathbb{Z}\)-lattices with determinant an odd prime \(p\) (or \(2p\) if \(p\) is not possible) are called primal lattices. It is proved that every isometry is generated by \(-1\) and symmetries with respect to minimal vectors of length 2 for primal even (or odd) \(\mathbb{Z}\)-lattices of rank less than 16 (or 13, respectively), provided that the minimal length is at least 2. Furthermore, it is shown that the restriction for the rank is sharp.