The authors provide a new approach to the classical result according to which the only possible isometric embeddings \(\ell_q^m\to\ell_p^n\) are those where \(q=2\), \(p\) is an even integer, and \(n\geq N(m,p)\), or where the underlying field is \(\mathbb R\) and \(\{p,q\}=\{1,\infty\}\). While this has been known for some time, the novelty in the new approach is to be largely field independent, and allows the theorem to be proved no matter if the underlying field is \(\mathbb R\), \(\mathbb C\), or \(\mathbb H\) (quaternions, in which case care has to be used to define the linear space, due to non-commutativity issues). The authors also prove estimates for the smallest dimension \(n\) for which (when possible) such embeddings do exist, and establish connections with cubature formulas for polynomial functions on projective spaces.