If you check the quaternion product derivation at wikipedia:
http://en.wikipedia.org/wiki/Quaternion#Hamilton_product
You can see that it is derived from a multiplication table between the quaternions 1,i,j,k. All books I have on the topic do the same. But the derivation assumes that quaternion multiplication distributes over quaternion addition (or am I wrong?). Is this a "valid" assumption, that does not need to be proven?