We won't make large intervention like in Iraq alone. But we can do regional ones (like in Ivory Coast). We did this with the backing of EU and UHN, to get as much legitimacy as possible.I agree; but France almost certainly would not act alone, without consulting at least the German government. While France wields some power - both politically and militarily - in its former African colonies, mostly, the French government seems to know the wisdom of consulting with allies concerning important geopolitical decisions. I don't think France would embark on a nearly unilateral adventure like the War in Iraq on its own.
But for a specific situation, the French president can send the French army on his own decision if needed. It doesn't mean he'll do that, but in case of emergency he has the power to do it.