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Another adviser and surge advocate who met with Mr. Bush in 2006 is Gen. Keane, at one time the Army’s vice chief of staff. Gen. Keane has bemoaned Iraq’s deteriorating security and political situation since U.S. troops left.

After the pullout was announced, he said: “We should be staying there to strengthen that democracy, to let them get the kind of political gains they need to get and keep the Iranians away from strangling that country. That should be our objective, and we are walking away from that objective.”

Analysts at the American Enterprise Institute also worked on and promoted the surge in 2006. Among them was Mr. Kagan, who writes in the Weekly Standard this week that the Obama administration is making false claims about Baghdad’s stability “even as Iraq collapses.”

“The withdrawal of all American military forces has greatly reduced America’s leverage in Iraq,” he wrote with wife and co-author Kimberly Kagan, who heads the Institute for the Study of War.

“U.S. military forces were a buffer to prevent political and ethno-sectarian friction from becoming violent by guaranteeing Maliki against a Sunni coup d’etat and guaranteeing the Sunnis against a Shiite campaign of militarized repression,” the Kagans wrote.

“The withdrawal of that buffer precipitated this crisis and removed much of our leverage. The withdrawal is complete and unlikely to be reversed. Still, the United States maintains some leverage in Iraq and considerable leverage in the region. The Obama administration will have to use all of its skills to maximize the impact of what leverage it retains.”

U.S. troops left on a timeline negotiated by the Bush administration before Iraq’s 2010 parliamentary elections. The idea was to give Mr. al-Maliki political cover before nationwide voting, then try to forge a new status of forces agreement in 2011 that would keep a stability force in place.

Critics say President Obama was so eager to leave Iraq that his administration made little effort to reach a new deal.

Mr. Biddle said the administration also assumed a hands-off approach as Mr. al-Maliki took nine grueling months to form a new government.

The prime minister now is relying on a voting bloc of lawmakers loyal to anti-U.S. firebrand Muqtada al-Sadr, a Shiite cleric with close ties to Iran. This alliance made it difficult for Mr. al-Maliki to buck Mr. al-Sadr and move to keep some U.S. troops on Iraqi soil.

“If you talk privately to Iraqi officials, they all wanted a sizable U.S. continued presence,” Mr. Biddle said. “They were afraid they would get demagogued by the Sadrists if they said so publicly. Because the United States took a hands-off position during government-formation negotiations, the result is Maliki turned to the Sadrists and not to [the Sunnis].”

The Pentagon wanted to keep a residual stability force of 3,000 to 6,000 in Iraq. Defense Secretary Leon E. Panetta said he could not let troops remain as long as Iraq failed to give them immunity from prosecution in Iraqi courts.

Critics said the administration did not try hard enough to reach a deal.

“We’re prepared to continue to negotiate with the Iraqis. We’re prepared to try to meet whatever needs they have,” Mr. Panetta told a Senate panel in November.

As U.S. troops exited Iraq on Dec. 14, Mr. Obama marked the war’s end in a speech at Fort Bragg, N.C.: “Iraq is not a perfect place. It has many challenges ahead. But we’re leaving behind a sovereign, stable and self-reliant Iraq, with a representative government that was elected by its people. We’re building a new partnership between our nations. And we are ending a war not with a final battle, but with a final march toward home.”