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If I'm not mistaken, the FFL was implicated in a murder plot to assasinate the president of France(a few decades ago). So, you can see they really aren't a French controlled entity.
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That was a complicated and bad time for the Legion. It was also the climax of a longstanding grudge match between the Legion and Charles de Gaulle.
During WWII, I think only one or two regiments were part of the Free French forces under the leadership of de Gaulle, the rest were aligned with Vichy France. I think opposing Legion elements actually fought against each other in Syria during WWII. I know for sure Free Legion forces landed in Norway to fight the German occupiers.
De Gaulle really hated the Legion after WWII for not supporting him. He was an outspoken critic of them and called for their abolition for the rest of his career.
Anyway, de Gaulle came to power during the Algerian war for independence. France actually considered Algeria as much a part of France as Normandy, Alsace, etc. Communists and far leftists called for Algerian independence, but de Gaulle vowed never to abandon it.
Then, he turned around and surrendered Algeria to the rebels in 1963.
Now this was a very bitter thing for the Legion. They felt they had lost Vietnam due to political indifference back in France, and wanted to redeem themselves in Algeria. Not only that, but Algeria's defense was the Legion's #1 mission, not to mention home base. When their chief political enemy stabbed them in the back, there was another schism in Legion loyalties, like back in WWII.
There was an actual rebellion, with the commander of the 1ere parachute regiment pledging allegiance to the Legion first, not France. The rebellion was put down and that unit dissolved, but disaffected Legionaires who now had no home plotted to kill de Gaulle in revenge. That's the basis of the movie "Day of the Jackal".
After that, the Legion became sort of the red-headed stepchild of the French Army. But it was revived in the early 70's when its new mission as a rapid deployment force was outlined. It is part of the French Army, and very much under French control.