quaternion product distributivity If you check the quaternion product derivation at wikipedia: < 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?

The multiplication is defined, in fact, as the unique product which distributes along sums, behaves nicely with scalars and takes the correct values at the basic elements.

That this _is_ actually such a multiplication is a simple result in linear algebra, similar to thaht which says that a linear map need only be given on a basis of its domain.