THE letter bore the corporate tone of a C.E.O. resolving a turf dispute between two middle managers.
In formal prose and numbered lists, Ayman al-Zawahri, the leader of Al Qaeda, directed one of the group's affiliates in Syria to withdraw to Iraq and leave operations in Syria to someone else.
The response was unequivocal. Abu Bakr al-Baghdadi, the leader of the Islamic State in Iraq and Syria, or ISIS, declared that his fighters would remain in Syria "as long as we have a vein that pumps and an eye that blinks."
It was the first time in the history of the world's most notorious terrorist organization that one of the affiliates had publicly broken with the international leadership, and the news sent shock waves through the online forums where jihadists meet.
In no uncertain terms, ISIS had gone rogue. That split, in June, was a watershed moment in the vast decentralization of Al Qaeda and its ideology since 9/11.
As the power of the central leadership created by Osama bin Laden has declined, the vanguard of violent jihad has been taken up by an array of groups in a dozen countries across Africa and the Middle East, attacking Western interests in Algeria and Libya, training bombers in Yemen, seizing territory in Syria and Iraq, and gunning down shoppers in Kenya.
What links these groups, experts say, is no longer a centralized organization but a loose ideology that any group can appropriate and apply as it sees fit while gaining the mystique of a recognized brand name.
In short, Al Qaeda today is less a corporation than a vision driving a diverse spread of militant groups. "Al Qaeda is kind of a ready-made kit now," said William McCants, a scholar of militant Islam at the Brookings Institution.
"It is a portable ideology that is entirely fleshed out, with its own symbols and ways of mobilizing people and money to the cause. In many ways, you don't have to join the actual organization anymore to get those benefits."
For policy makers and terrorism analysts, this has made it harder to define what it means to be "Al Qaeda" and to gauge and combat threats.
In addition, disagreements over definitions of Al Qaeda have animated debates in Washington about the perpetrators of the 2012 attack on the American diplomatic compound in Benghazi, Libya, especially over the Benghazi militant group Ansar al-Shariah.
Although intelligence agencies and the State Department do not consider the group an affiliate of Al Qaeda, some Republican critics of President Obama argue that its puritanical, anti-Western vision makes it one.
In many ways, American counterterrorism operations since 9/11 have successfully handicapped the original Qaeda organization founded by Bin Laden from the remnants of the mujahedeen who fought against the Soviets in Afghanistan in the 1980s.
Drone strikes have eliminated top leaders, surveillance impedes communication with affiliates and the killing of Bin Laden removed a charismatic, unifying figure.
The franchise model has been essential to the group's survival, even if that means affiliated groups are often left to their own devices and focus more on local struggles than on attacking the West.
"There is really not one Al Qaeda anymore," said Gregory D. Johnsen, the author of "The Last Refuge," a book on Al Qaeda in Yemen. "It has taken on the local flavor of wherever it is, although none of the groups have really disavowed transnational jihad."
The conflict in Syria, currently the world's biggest draw for international jihadists, illustrates how difficult it can be to characterize militant groups. ISIS, despite having publicly rebuffed Mr. Zawahri, has thousands of foreign fighters who have seized territory in Iraq and Syria, where it seeks to found an Islamic state.
Its rush to control resources and impose strict Islamic codes caused a backlash from other rebel groups, and fighting between the sides has killed more than 1,400 people in recent weeks.
Syria's other Qaeda group, the Nusra Front, remains loyal to Mr. Zawahri, but has also maintained good relations with the wider rebel movement that shares the American goal of getting rid of President Bashar al-Assad.
Beyond its open affiliation with Al Qaeda, little separates the Nusra Front from other Islamist battalions fighting in Syria.
One of them, Ahrar al-Sham, even has a Qaeda member in its leadership. While these groups' Islamic vision for the future of Syria may disturb many Americans (and Syrians), they have not attacked Western targets.
"There are a lot of militant groups out there that are supporting either Islamist, salafist or jihadi doctrine, but they are not all wrapped up about the U.S., so do you call them all Al Qaeda?" said Clint Watts, a former F.B.I. agent now with the Philadelphia-based Foreign Policy Research Institute.
He and other analysts argue that branding all such groups Al Qaeda is counterproductive and say it makes more sense to differentiate between groups with a local focus and those actively fixed on attacking the West.
In this line of thinking, the most worrisome group is Al Qaeda in the Arabian Peninsula, which is led by an associate of Bin Laden's in Yemen and has repeatedly tried to attack the United States.
This makes it a much greater threat, Mr. Watts said, than Qaeda-inspired groups that have taken up arms against their governments in North Africa, or Boko Haram, which has waged a sectarian terrorism campaign in northern Nigeria.
Others argue that even localized groups see themselves as fighting an international struggle that they'll join when they can. Many assumed that the Qaeda-affiliated Shabab in Somalia was local until their gunmen stormed into the Westgate mall in Nairobi.
And the shadowy Algerian jihadist Mokhtar Belmokhtar seemed to be focused on his trans-Saharan smuggling and kidnapping business until his group raided a natural gas plant in In Amenas, Algeria, looking for foreigners to capture and kill.
"No Qaeda threat has ever remained exclusively local," said Bruce Hoffman, director of security studies at Georgetown University. "They have always eventually crossed borders and become regional in operations and attacks and certainly in fund-raising and recruiting."
Even groups with no direct links to Al Qaeda can seize opportunities to strike American targets, as in the Benghazi attack in 2012 that killed Ambassador J. Christopher Stevens. In that case, the American focus on Al Qaeda may have contributed to the failure to foresee other dangers brewing nearby.
But while counterterrorism can be effective in stopping specific threats, depriving militant groups of the unstable environments where they flourish and organize is much harder.
"What you are seeing in the Middle East is a problem of militancy combined with ungoverned territory," said Ramzy Mardini, a fellow at the Atlantic Council. "That is the real problem, not which groups belong to Al Qaeda and how can we get rid of them."
By Ben Hubbard