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Gen. Keane, on the other hand, told Gen. Petraeus that he had entered the select pantheon of famous U.S. generals, in the model of Gen. Dwight Eisenhower.

“We haven’t had a general like you in a long time,” Gen. Keane said, telling him there were only two posts in the military that he should accept after his tour in Iraq: head of CentCom or Chairman of the Joint Chiefs.

Other than that, Gen. Keane hinted, there was a possible future in politics.

Gen. Petraeus will hand over control of the U.S. military operation in Iraq on Sept. 16 to Gen. Raymond Odierno, and take control of CentCom. He has said he has no political plans.

“The War Within” is Mr. Woodward’s fourth book on the Bush White House and its response to the terrorist attacks on Sept. 11, 2001. It is the reporter’s 14th book in 36 years since helping uncover the Watergate scandal inside the Nixon administration.

The book tells the story of how the Bush administration decided to send a “surge” of 30,000 more troops into Iraq during the spring of 2007, after violence in 2006 spiraled out of control.

The book credits National Security Advisor Stephen Hadley with driving a secret strategy review process in late 2006, though Mr. Woodward faults Mr. Bush for being too disengaged from the process.

Mr. Woodward also says that Mr. Hadley was too often a “cheerleader for [Mr. Bush’s] greatness,” and that he failed, prior to the 2006 strategy review, to implement a thorough, deliberate policy-making process, instead relying too heavily on the president’s instincts.

Mr. Hadley on Friday issued a response to Mr. Woodward’s book, saying that the president was not “detached” from the review process, but instead “drove the process to conclusion and made a tough decision.”

Violence in Iraq has declined dramatically since the surge, and political progress among competing Iraqi factions, while moving at a much slower pace, has occurred.

Mr. Woodward’s book argues that reductions in violence are due in part to the surge, but that three other factors have been as or more important.

First, the U.S. military has used new, highly classified and advanced techniques to kill or capture insurgent leaders. Second, in fall 2006 Sunni sheiks west of Baghdad chose to side with the U.S. against Al Qaeda.

And third, in August 2007, powerful Shiite cleric Moqtada al-Sadr suspended the activities of his militia.

Mr. Hadley also debated this point.

“It was the President’s decision in January 2007 to ‘surge’ an additional 30,000 troops into Iraq that ‘enabled’ the other three factors,” Mr. Hadley said.