We don't have anything final and concrete, but we know for sure that Earth Alliance is actually founded by Federation of Earth [or Earther Federation] as a result of a war against Martian Hegemony they won. The name is a dire reminder for martians that they lost the war and should always remember about their position within the Alliance. There was probably a war against TransJovian Hegemony, but due to decentralized nature of rim governments, it ended up with stronger despotes willingly join the EA for personal benefit of their leaders.
I feel that Earther government that become dominant power within the EA was actually elected, but subsequently dismantled the post-GTA democratic institutions and established autoritatian regime instead. They are legitimate solar government, even if they don't respect the rule of law and isonomia. It was basis of their territorial claims for the rest of terran colonies outside Sol, after reopening of subspace corridor between Sol and Delta Serpentis.
It's worth noting that certain military leaders, such as Admiral Falon have very high dose of inidividual power and influence inside EA structures, so despite the fact that they do have a parliment, EA is closer to military oligarchy.