Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$.  I am interested in finite cycles (i.e.,
the sequence eventually returns to the original $A$ and $B$).  Is this studied
somewhere?  What is known about the maximum order of such cycles?  (By the way, it
is easy to see that my original "non-commuting" requirement can be relaxed to
"non-equal" because I am dealing with a cycle.)

In particular, I've actually been looking at quaternions and one example cycle
(with order 8) can be generated from:
$$A=\space\space\frac{1}{\sqrt{2}}-\frac{j}{2\sqrt{2+\sqrt{2}}} + \frac{k}{2\sqrt{2-\sqrt{2}}}$$
$$B=\space\space\frac{1}{\sqrt{2}}+\frac{j}{2\sqrt{2+\sqrt{2}}} + \frac{k}{2\sqrt{2-\sqrt{2}}}$$
I can find longer cycles, but only numerically.  So, in the space of just quaternions, what is the largest cycle order possible?
Is it infinite?