In 2003, he commanded the 101st Airborne Division in Operation Iraqi Freedom, which after the incursion into Iraq took up station in Mosul in Kurdistan. There he made a hallmark radical policy decision, ordering his men to get among the people and resurrect schools, law and government. Some criticised his choice of local officials, but the experiment worked. His next command was to sort out government in Baghdad itself, a less happy venture, until the Anbar Awakening movement began to counter the Sunni insurgency. 

When Petraeus took over the running of Iraq, he succeeded not only by the invention of the surge, but a particularly acute choice of colleagues and deputies. One of his closest collaborators and allies has been Lieutenant-General Sir Graeme Lamb, who conducted crucial covert negotiations with Sunni leaders for Petraeus, and his Special Forces commander in Iraq, Stanley McChrystal. Petraeus also brought in advisers such as the now retired Lieutenant-Colonel John A. Nagl, the author of Learning to Eat Soup with a Knife, and the Australian David Kilcullen, whose The Accidental Guerrilla is an equally seminal work on informal warfare. 

Their thinking has been crystallised in The US Army and Marine Corps' Counterinsurgency Manual (Field Manual 3-24), a good example of a ground-breaking work with a thoroughly dull title. "A counterinsurgency campaign is...a mix of offensive, defensive and stability operations conducted along multiple lines of operations," Petraeus states in his preface of 2006. "It requires Soldiers and Marines to employ a mix of familiar combat tasks and skills more often associated with non-military agencies." 

In his present post as Commander, Central Command, Petraeus has direct command of only the counter-terrorist Operation Enduring Freedom in Afghanistan. But his men and ideas are everywhere there, and he advises President Obama on the Afghan mission.

The only British near-equivalent to Petraeus is General Sir Rupert Smith, another charismatic intellectual commander. His ideas on the "utility of force" and "wars among the people" are now taught in the courses Petraeus inspired. But in their personal approach to power and influence they diverge. Smith remains the free-thinking independent outsider. Petraeus is the wily establishment insider, which is why it has been suggested he will run for President, probably in 2016.

He has officially denied presidential ambitions. But when I asked a British major-general who had worked happily for Petraeus in Baghdad whether the White House was off the agenda, he said, "Don't count on it."