Why and how are quaternions 'bilinear'?

Question

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as:

While others specify:

Excuse the poor images, StackExchange did not seem to like my LaTeX.

Note that the sign on the vector cross product has been altered. I think this is a result of the bilinear property of quaternions, but I am not sure.

If this is the case, which is the 'true' quaternion multiplication? Is it simply a matter of convention? What alterations to the properties of quaternions take place if I choose one over the other?

My understanding was that quaternions offered a 'unique' representation of rotations in R3. Is this not impacted by the existence of two multiplication operators?

My background is Applied Math/Engineering.

Answer 1

bilinear: Bilinear over the reals. Linear in each variable separately. (1) Linear in the first variable: $$ (tp)q = t(pq),\qquad (p_1+p_2)q = (p_1q)+(p_2q) $$ for all quaternions $p, p_1, p_2, q$ and all reals $t$. (2) Linear in the second variable (left to the reader).

....

which is the 'true' quaternion multiplication?

Well, Hamilton said $ij=k$, so I guess that is your second one.