The Quaternions are the smallest.... I've been reading https://github.com/GleasSpty/MATH-104-----Introduction-to-Analysis, and the author formulates the integers as the smallest (by inclusion under isomorphism) nontrivial totally ordered cring that contains the natural numbers, the rationals as the smallest totally ordered field that contains the integers, and the reals as the smallest dedekind-complete (or cauchy-complete) totally ordered field that contains the rationals. Similarly, there's the algebraic numbers which are the smallest (edit: they're not totally ordered) algebraically complete field that contains the rationals, and the complex numbers which are both algebraically complete and dedekind-complete.
Is there a similar statement for the Quaternions/Octonions?

By Frobenius' theorem, the quaternions $\Bbb{H}$ can be characterized as the smallest noncommutative division ring that contains $\Bbb{C}$.
