 Finite-dimensional real division-algebra classification

A finite-dimensional associative division algebra over $\mathbb{R}$ is one of the three displayed algebras. A nonreal element has an irreducible quadratic minimal polynomial and yields $i^2=-1$. Its centralizer is a division algebra over central $\mathbb{C}$, hence is $\mathbb{C}$ by splitting minimal polynomials. Decompose the whole algebra into the plus and minus eigenspaces of conjugation by $i$. If the minus space is zero, the algebra is $\mathbb{C}$; otherwise it is $\mathbb{C}j$, since multiplying by $j^{-1}$ takes that space into the centralizer. One has $jz=\overline{z}j$ and $j^2\in \mathbb{R}$. Division forces $j^2<0$, which can be rescaled to $-1$, yielding $\mathbb{H}$. Of these, only $\mathbb{R},\mathbb{H}$ have center $\mathbb{R}$.