Sorry, I had to get that off my chest. The Falange was one right-wing group among many others at the beginning of the Civil War. Franco was not a member of it, though later he folded it back into his own movimiento in order to control it. Franco, ideologically, was a Monarchist, although he never restored the monarchy except by a posthumously executed decree.
The man who arrested Lorca, Ramón Ruiz Alonso, was a Falange reject, and Lorca was hiding in the home of his own Falangist friends, the Rosales family. It could have been Ruiz Alonso's resentment against the Falange that helped to get Lorca killed.
I have no love for the Falange, a Spanish Fascist organization founded by José Antonio Primo de Rivera, the son of a dictator in power in the 20s. I do have a certain love, or respect, for historical accuracy. Not even Franco killed Lorca, if we must be precise. He was not the general in charge of Andalusia, at the time.
Of course, later on, in the war, the entire conflict was seen internationally and to some extent internally as one between communists and fascists. At the beginning of the war, though, there were numerous political factions on the right, the left, and the center. Falangists, Carlists, Anarchists, Socialists, Communists, other Monarchists, and even some liberal Republicans.