I have been reading about the Hurwitz quaternions $ H $, and parallel to reading, trying to perform some of the basic operations on some examples.
There are plenty of places online where the division algorithm is stated for the Hurwitz quaternions, which I will restate here:
Given $ a, b \in H $, there exists $ q, r \in H $ so that $ a = bq + r $, where for the norm $ N(x) $, $ N(r) < N(b) $
Dividing out terms on the left this way, is there an easy way to actually construct what q is in this example? The lattice argument for why such $ q, r $ must exist makes some sense to me, but I don't know how to actually compute $ q \in H $ (if I can compute one, I can compute the other too).
I did find references to the inverse being defined as the conjugate over the norm, and this can be used to compute the quotient if $ b \space \vert \space a $. However, I'm not sure how to perform the standard division algorithm otherwise on Hurwitz quaternions, as this is usually a given in the integers or left up to the computer scientists.
I also really don't want to simply subtract one value from the other repeatedly, as that seems slow, and I haven't figured out if that works, especially if two whole-integer quaternions could have a half-integer remainder, which I thought must happen because of the failings of the Lipschitz quaternions.
So to summarize, is there any easy way to express $ q $ in terms of integer quotients on the components of $a$ and $b$?
Wrtie $ab^{-1}=t_1+t_2i+t_3j+t_4k$, with $t_1,t_2,t_3,t_4$ rational. Now every real number $t$ satisfies $E(t+1/2)\leq t\leq E(t)+1/2$.
If at least two of the $t_i$'s satisfy $\vert E(t_i+1/2)-t_i\vert\leq 1/4$, set $q=E(t_1+1/2)+E(t_2+1/2)i+E(t_3+1/2)j+E(t_4+1/2)k$.
If not, at least two of the $t_i$'s satisfy $\vert E(t_i)+1/2-t_i\vert\leq 1/4$. Then set $q=E(t_1)+1/2+(E(t_2)+1/2)i+(E(t_3)+1/2)j+(E(t_4)+1/2)k$.
In both cases, you can check that $N(ab^{-1}-q)\leq 5/8$. Then set $r=a-bq$.