Quaternions - Generalizations

Generalizations

If F is any field with characteristic different from 2, and a and b are elements of F, one may define a four-dimensional unitary associative algebra over F with basis 1, i, j, and ij, where i2 = a, j2 = b and ij = −ji (so (ij)2 = −ab). These algebras are called quaternion algebras and are isomorphic to the algebra of 2×2 matrices over F or form division algebras over F, depending on the choice of a and b.