Metaconjugation and Quadratic Forms

We present a quaternionic proof that the quadratic form $t^2+2x^2+5y^2+10z^2$ represents all positive integers, and that the form $t^2+x^2+7y^2+7z^2$ represents all natural numbers which are neither equal to $3\cdot 7^{\mathrm{\ell }}$ nor equal to $6\cdot 7^{\mathrm{\ell }}$, for any $\mathrm{\ell }\in {\mathbb{N}}_0$. To do this we introduce a technique that may be useful for other purposes, and that we call “metaconjugation”.