Real matrix representations of quaternions

The quaternions were invented in 1843 by Hamilton. There are three quaternions: i, j and k where i is the well known imaginary number i. The rules for the multiplication of quaternions are as follows :

i * j = k, j * i = -k
j * k = i, k * j = -i
k * i = j, i * k = -j
i^2 = j^2 = k^2 = -1.

The quaternion group has eight members { 1, -1, i, j, k, -i, -j, -k } and is non-abelian.

The most common presentation of Q8 is < a,b | a^4 = 1, a^2 = b^2, b^-1*a*b = a^-1 >.

Using only real numbers the quaternions can be represented as 4-by-4 matrices as follows.