The US didn't start the Lybia thing, France (and particularly Sarkozy) did, and the US felt obligated to follow up. In Hillary leaked emails, you can read about it. The CIA basically saying 'the successor France want is X, let's try to put our choice as the leader of the next Lybia instead', and a week later, the french/DGSE candidate for Khadafi succession died.

Lybia was 100% a French war the US took over. Sarkozy's subordinates were extremely close to the Lybia regime that helped illegally finance his presidential election, at least according to French judges, so it was also also a very political war.