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

1.
The real numbers $\mathbb{R}$.

2.
The complex numbers^{} $\u2102$.

3.
The Hamiltonians (also known as the quaternions) $\mathbb{H}$.

4.
The octonions (also known as the Cayley or CayleyDickson 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 crossproduct 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.
Title  composition algebras over $\mathbb{R}$ 

Canonical name  CompositionAlgebrasOvermathbbR 
Date of creation  20130322 17:18:17 
Last modified on  20130322 17:18:17 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  7 
Author  Algeboy (12884) 
Entry type  Example 
Classification  msc 17A75 
Related topic  HurwitzsTheorem 
Related topic  JacobsonsTheoremOnCompositionAlgebras 