Let us call "division algebra over $\mathbb R$" a finite-dimensional vector space $A$ equipped with a bilinear map $A \times A \to A: (a,b) \mapsto a \bullet b$ , such that $a\bullet b=0$ implies $a=0$ or $b=0$. ( Associativity is not required).

Examples : the reals, the complexes, the real quaternions and the octonions of Graves-Cayley.

Any such division algebra must necessarily have dimension 1,2,4 or 8 (as in the examples). This was proved indepently in 1958 by Kervaire and Milnor using Bott's periodicity theorem, a fantastic result in algebraic topolgy which had just been proved.

To the best of my knowledge there is still no purely algebraic proof of this theorem on possible dimensions of real division algebras, although the statement is completely algebraic and elementary.