Complex sentence: Thereafter, Chávez began to lead a "double life": while feigning disciplined obedience amidst his military superiors and miming political neutrality around his family, Chávez engaged in secret trysts with known socialists, communists, and other leftists.
Split into simpler sentences: Thereafter, Chávez began to lead a "double life": he feigned discipline and obedience around his superiors in the military. Discreetly, however, Chávez engaged in secret trysts with known socialists, communists, and other leftists.