Doing the matrix multiplication explicitly gives:




From which the required commutation rules follow immediately.
We can relate the L's to quaternions by mapping:

Note that the bold
i here is the quaternion
i, whereas the non-bold
i multiplying the L's is just the complex number
i.
This is the mapping corresponding to

, etc. Another, equally good mapping can be obtained if we drop the minus signs, corresponding to quaternion rotations with the direction of rotation reversed.