Why did the Teradaya Incident, a tragic incident in which Satsuma samurai fought among themselves, occur? What was the outcome?

The "XX-ya Incident" in Kyoto always comes up in works about the end of the Edo period. "Where was Ryoma Sakamoto assassinated?" "The Shinsengumi also rampaged, right?" "Wasn't Satsuma involved?" Well, the combination of "hotel name + incident" from that time is just confusing. In particular, the Teradaya Incident, which we will be focusing on this time, has two incidents with the same name, so it can be even more difficult for those who are not…