According to Dickson we call a positive definite integral quadratic form \(f\) on \(\mathbb{Z}^ n\) regular, if for every positive integer \(a\) a local representation of \(a\) by \(f\) at all completions of \(\mathbb{Z}^ n\) implies global representation of \(a\) by \(f\). In the paper under review the author studies a higher-dimensional analogue of Dickson's regularity condition. Let \(R= \mathbb{Z}\), \(F=\mathbb{Q}\) or \(R=\mathbb{Z}_ p\), \(F=\mathbb{Q}_ p\) (the \(p\)-adic completion of \(\mathbb{Z}\) resp. \(\mathbb{Q}\)). Let \(V\), \(W\) be quadratic spaces over \(F\). An \(R\)-lattice \(K\) of \(W\) is said to be represented by the \(R\)- lattice \(L\) of \(V\), if there exists an isometry \(\sigma: W\to V\) such that \(\sigma(K) \subseteq L\). Now let \(L\) be a quadratic \(\mathbb{Z}\)-lattice of rank \(n\) and let \(k\leq n\) be a positive integer. Then \(L\) is said to be \(k\)-regular if \(L\) represents all quadratic \(\mathbb{Z}\)-lattices \(K\) of rank \(k\) for which \(L_ p\) represents \(K_ p\) for all prime spots \(p\) of \(\mathbb{Q}\). For \(k=1\) this is the definition of regular given by Dickson. The main result is that there are only finitely many inequivalent 2-regular primitive positive definite quadratic lattices of rank 4. It is already known that this result is false if we replace 2-regular by 1-regular. The proof of the main theorem is based on a detailed analysis of the successive minima \(\mu_ 1\), \(\mu_ 2\), \(\mu_ 3\), \(\mu_ 4\) of the quaternary lattice \(L\). These are the positive integers defined by \(\dim (\text{span} \{x\in L, f(x)\leq\mu_ j\})\geq j\) and \(\dim (\text{span} \{x\in L; f(x)< \mu_ j\})