There are 7 non-isomorphic composition algebras over $\mathbb{R}$ , first 4 division algebras and secondly 3 split algebras.

1. The real numbers $\mathbb{R}$ .
2. The complex numbers $\mathbb{C}$ .
3. The Hamiltonians (also known as the quaternions) $\mathbb{H}$ .
4. The octonions (also known as the Cayley or Cayley-Dickson algebra) $\mathbb{O}$ .
5. The exchange algebra: $\mathbb{R}\oplus\mathbb{R}$ .
6. $2\times 2$ matrices over $\mathbb{R}$ : $M_2(\mathbb{R})$ .
7. The cross-product of $2\times 2$ -matrices over $\mathbb{R}$ : $M_2(\mathbb{R})\circ M_2(\mathbb{R})$ .

The proof can be seen as a consquence of a theorem of Hurwitz and a theorem of Jacobson. In reality various authors contributed to the solution including Albert, Dickson and Kaplansky.