The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below).
Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic.
The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras.
In general by theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of K of degree equal to the index of A, and this splitting field is isomorphic to a subfield of A.
[12][13] As an example, the field C splits the quaternion algebra H over R with We can use the existence of the splitting field to define reduced norm and reduced trace for a CSA A.