First Triumvirate Triumvir or tresvir: member of a college of three members. The expression is mostly used to describe the First Triumvirate (60 BCE; Pompey the Great, Crassus, and Julius Caesar) and Second Triumvirate (43 BCE; Marc Antony, Lepidus, and Octavian).
With 8 nights, I suggest you spend 3-4 in Tokyo, 1 in Hakone for ryokan, 3-4 in Kyoto. Hakone offers Mt. Fuji views on a clear day, and it sits right off the train line between Tokyo and Kyoto so it's an easy detour. While at Kyoto, spend a day at Nara.
While in Osaka/Kyoto i would take a day trip to Hiroshima & Miyajima Island and a day trip to Nara as well. Or, i would also do a 4 nights stay in Osaka/Kyoto, 2 nights stay in Fuji and 1 week in Tokyo. It really depends on whats your preference.