Let F be a field. We will construct the generalized quaternions and the generalized octonions over F, respectively. This construction will depend on the three parameters a,b,c in the field F.
First, let a be an element of F with for each u in F. Then the multiplication
turns into a field. This field becomes a Dickson-algebra using the conjugation Of course, has dimension 2 over F.
Now, let b be another element in F with for all elements u1, u2 in F. Let be the Cayley-Dickson extension of with respect to the given parameter b. Then becomes a non-commutative skewfield, with F as its center. This skewfield is called a generalized quaternion algebra over F, obviously it is of dimension 4 over F.
Finally, let c be another parameter in F with for all u1, u2, u3, u4 in F. The Cayley-Dickson extension of with respect to the parameter c is called a generalized octionon algebra over F. Note that is a proper alternative field (i.e. is not associative,) of dimension 8 over
The classical quaternions and octonions are obtained by using the field of real numbers as our base field F with the parameters a=b=c=-1.