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.