Barack Obama has stated that the U.S. is at war with al-Qaeda. I suggest that the interested reader search google on "U.S. supports al qaeda in libya" and then on "U.S. supports al qaeda in Syria". These will lead one into articles verifying that the U.S. has provided material support to terrorists that call themselves al-Qaeda or otherwise link themselves to this term, whatever organization of umbrella beliefs it stands for.
How can the U.S. be at war with al-Qaeda and be providing material support to it in overthrowing governments in Libya and Syria? It cannot. How can the U.S. be at war with al-Qaeda and maintain Saudi Arabia as a close ally, when that country is a major source of al-Qaeda people and teaching?
The idea put forward by two administrations that the U.S. is at war with al-Qaeda is a "Big Lie" that is designed to provide cover for other motives for their military actions. It is a very simple and convenient cover story that sells to the public as a rationale for all sorts of U.S. military activities in many countries, all of which have other explanations.
Attacking Iraq clearly had nothing to do with al-Qaeda. The Bush administration made a feeble attempt to link some al-Qaeda operatives to time spent passing through Iraq, but they gave it up. Instead they substituted a different Big Lie, which was weapons of mass destruction (WMD) and imminent threats thereof.
Attacking the government of Afghanistan had nothing to do with al-Qaeda, and to this day the military and other activities in Afghanistan have nothing to do with al-Qaeda. The government of Afghanistan asked, quite reasonably, for evidence of al-Qaeda participation in the 9/11 occurrences. The U.S. didn't provide it.
The spread of U.S. drone warfare into Pakistan and Yemen has little or nothing to do with al-Qaeda, even if the U.S. happens to attack those who may or may not identify themselves with al-Qaeda. It is clear that the U.S. has other geopolitical aims in extending its warfare. It is clear that certain beneficiaries of this extension are having influence in Washington's policies, and this has nothing to do with al-Qaeda.
If Washington actually regarded the nation as at war with al-Qaeda, would it undertake policies that swelled the ranks of the designated enemy? But that is exactly what it is doing, and reports to that effect are now appearing in Washington. This can be understood if Washington's hidden and deeper aims are something other than war on al-Qaeda.