Quaternions as pairs of complex numbers

Quaternions can be represented as pairs of circuitous numbers. From this perspective, quaternions are the aftereffect of applying the Cayley–Dickson architecture to the circuitous numbers. This is a generalization of the architecture of the circuitous numbers as pairs of absolute numbers.

Let C2 be a two-dimensional agent amplitude over the circuitous numbers. Choose a base consisting of two elements 1 and j. A agent in C2 can be accounting in agreement of the base elements 1 and j as

If we ascertain j2 = −1 and ij = −ji, again we can accumulate two vectors application the distributive law. Writing k in abode of the artefact ij leads to the aforementioned rules for multiplication as the accepted quaternions. Therefore the aloft agent of circuitous numbers corresponds to the quaternion a + bi + cj + dk. If we address the elements of C2 as ordered pairs and quaternions as quadruples, again the accord is