Isometric embeddings of finite-dimensional $\ell_p$-spaces over the quaternions

The nonexistence of isometric embeddings $\ell_{q}^{m}\to\ell_p^{n}$with $p\neq q$ is proved. The only exception is $q=2$, $p\in 2\mathbb{N}$, in which case an isometric embedding exists if $n$ is sufficiently large, $n\ge N(m,p)$. Some lower bounds for $N(m,p)$ are obtained by using the equivalence between the isometric embeddings in question and the cubature formulas for polynomial functions on projective spaces. Even though only the quaternion case is new, the exposition treats the real, complex, and quaternion cases simultaneously.