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Sep 26, 2017 at 15:44 comment added Ben McKay This doesn't answer the question quite. You can see that your $f$ generates a finite abelian group, since some power fixes 4 points. So then $f$ generates a finite cyclic subgroup of $PU(3)$, being compact so lying in a maximal compact. One can then diagonalize $f$. I hope this helps.
Sep 26, 2017 at 15:36 history edited Ben McKay CC BY-SA 3.0
added better reference
Sep 26, 2017 at 15:26 history answered Ben McKay CC BY-SA 3.0