Technically that was a different chapter of the BoS, fallout 4 went back to the West, if I remember correctly
Minor Spoilers: [
Elder Lyons died after FO3, and soon after Sarah Lyons died in combat, leaving no Elder. There was a leadership crCIA and several leaders were elected and quickly deposed. The West Coast Elders, now back in contact, elected Maxson at the age of 16, just 4 years before Fallout 4. He was given the position for defeating a Super Mutant leader, and brokering a peace with the Brotherhood Outcast.].
Maxson's leadership is much more in line with the traditional view to collect/preserve technology, and to wipe out the abominations caused by the abuse of technology (mutants and now synths). Yet he still maintains some care for the people of the Wasteland, but not enough to stop the quest for tech, as Lyons did.
However, that all comes from the Fallout wiki, and has no source.
I didn't encounter any of the info about his election over the course of the game.