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The White House cheered the news Thursday of an agreement that ended an eight-month deadlock in forming a new government in Iraq.

Re-elected Iraqi President Jalal Talabani named Shiite Nouri al-Maliki to stay on as prime minister and to assemble the next government after a dramatic walkout by members of a Sunni-backed coalition.

White House officials said the U.S. has been engaged but called this a government "made in Iraq."

"What we've seen is an emergence of politics," a senior administration official said. "This has been a difficult process and all sides had to compromise but they are dealing with important issues."

Al-Maliki now has 30 days with which to form a new government.

His efforts to include significant Sunni support in the new government took a beating after the walkout by members of the Sunni-backed Iraqiya bloc. Iraqiya lawmakers had demanded that before the parliament votes on the president, it vote first to formally reverse decisions by a so-called de-Baathification program barring three of their members from holding government positions.

The U.S. praised the fact that the new government would have at least some Sunni presence.

"The apparent agreement to form an inclusive government is a big step forward for Iraq," said Tony Blinken, national security adviser to Vice President Biden, the administration's point man for Iraq.

The power-sharing deal reached Wednesday night was heralded by some politicians as a breakthrough, ending the months of wrangling since the inconclusive March 7 parliament elections. But Sunnis were already accusing al-Maliki of not fulfilling promises and have warned they could pull out if they are not met.

At a press conference after the walkout, a lawmaker from Iraqiya, Haider al-Mulla, did not answer when asked whether the bloc would participate in the government. Instead, he said Iraqiya would seek "explanations from al-Maliki and State of Law over their broken commitments." State of Law is another

The first vote went smoothly, with a Sunni from Iraqiya, Osama al-Nujaifi, elected parliament speaker. But before the vote to elect a president, 57 Iraqiya lawmakers walked out.

Iraqiya lawmakers have said that as part of the power-sharing deal, the other factions agreed to get rid of the controversial De-Baathification law entirely within two years. Sunnis view the De-Baathification process as a thinly veiled Shiite attempt to bar Sunnis from returning to power.

Despite the walkout -- which Iraqiya leader Ayad Allawi and the parliament speaker joined -- lawmakers still had quorum and proceeded with the session. They elected Talabani as president for a second term. Talabani addressed the lawmakers remaining in the parliament hall, declaring, "Today is a day of victory, the victory of the free Iraqi will."

He then took the constitutionally required step of formally requesting that al-Maliki form a new government. Al-Maliki has 30 days to do so, as the factions work out the allotment of ministry positions, including key posts like foreign affairs and the interior ministry in charge of security forces.

Even if the power-sharing deal holds, it could potentially be a setback for the U.S., which had been pushing for a greater Sunni say in power, and a boost for regional rival Iran.

The Sunni minority had put great hopes in the March elections and succeeded in lifting their bloc to a narrow victory: Allawi's Sunni-backed Iraqiya coalition won the most seats in the March elections, but not a majority. Al-Maliki, whose State of Law party came in second, cobbled together alliances with Iranian-backed religious Shiite parties, gathering enough seats to thwart Allawi's bids for either the prime minister job and the presidency.

Instead, under the power-sharing agreement, Allawi is to lead a newly created council to oversee issues of security and foreign policy. But the council's powers remain vague, and already there were signs of a fight brewing over the extent of its authority.

The extent of Sunni power in the new government will not become clear until the allocation of ministry posts is announced. But their weight in the government will depend on other factors as well. Power in Iraq is often determined not just by formal roles but by the personalities of the individuals and their party backing.

The presidency, for example, is largely a ceremonial job, but Talabani has been able to wield considerable power because of his background as a longtime Kurdish leader and the Kurds' dominance in the north. In contrast, an earlier Sunni defense minister was an independent with no party backing and less authority that his title would suggest.

Allawi has been criticized for his lack of participation in the previous parliament, preferring instead to spend time at his London house. For his council to have any weight, Allawi would have to play a vigorous and daily role.

The new government could also give a significant role to the Sadrist movement, led by anti-American cleric Muqtada al-Sadr, who has grown closer to Iran. Al-Maliki's alliance with the Sadrists was key to keeping him in power.

Fox News' Jake Gibson and The Associated Press contributed to this report.