Minimum number of real multiplications to multiply two quaternions Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows:
$$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$
We only need the products $ac$, $bd$ and $(a+b)(c+d)$. Does there exists a similar trick for quaternion multiplication? A naive multiplication would need $16$. This can probably be reduce to $9$ using the same trick as above (and taking into account that quaternion multiplication is non-commutative). My question is it known that $9$ is the lower bound? If not, does there exist an algorithm, which does quaternion multiplications in less than $9$ real multiplications?
de Groote showed that one could compute both $uv$ and $vu$ using only a total of 10 non-scalar multiplications.

Howell and Lafon, in a tech report (that I could not find a refereed version of and have not checked myself, but is probably correct) showed that one can do quaternion product with only 8 multiplications. 

7 is certainly a lower bound, as noted in both of the above papers.

(While this was an interesting question and fun to look up, I found the answer in about 5 minutes of Googling. Google is your friend.)