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?
EDIT (Aug 10 2021): Without loss of generality for the quaternion problem, I believe this is just a matter of finding finite cycles which can be generated from two degrees of freedom (real $\alpha$ and $\beta$): $$A=\space\space\sqrt{1-\alpha^2-\beta^2}-\alpha{j} + \beta{k}$$ $$B=\space\space\sqrt{1-\alpha^2-\beta^2}+\alpha{j} + \beta{k}$$ Seems easy, doesn't it?
