Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 10 / a / Solution

The explicit matrices areDirect matrix multiplication givesThe reverse products have the opposite signs: $YX=\hat{Z}$, $ZY=\hat{X}$, $XZ=\hat{Y}$. Since $W$ is the identity matrix and $X^2=Y^2=Z^2=-W$,These are the usual quaternion multiplication relations in the complex matrix representation of quaternions.