"A similar operation will be launched in al-Ghazaliya against al-Qaeda today. We have sufficient information on places they are in, and we will punish them.
Al-Hais said his forces were fighting in plain clothes.
The Salvation Council is the armed wing of an alliance of Sunni sheikhs from the western Iraqi province of Anbar, who have encouraged men to the Iraqi security forces in order to fight al-Qaeda in Iraq members.
Many of these Sunni fighters were previously hostile to the US military and Baghdad's Shia-led government but, angered by al-Qaeda's attacks on civilians and tribal leaders, they have now changed sides.
US commanders see this as one of the most positive recent developments in Iraq, which is in the grip of vicious series of overlapping conflicts, and hope now to persuade armed groups to join a peace process.
"Let the people of Iraq vote if they want the US to stay or leave"
Bob Kaye, Bohemia, US
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Lieutenant-General Raymond Odierno, the number two US officer in Iraq, said on Thursday that about four-fifths of the groups currently fighting American forces were thought to be ready to end their campaigns.
"So we want to reach back to them," he said. "And we're talking about ceasefires and maybe signing some things that say they won't conduct operations against the government of Iraq or against coalition forces."
Amiriya, however, was still far from peaceful as the political shifts played themselves out in the form of bloody street battles.
Official casualty figures were not immediately available.
But a local council member, who declined to be identified because of security concerns, said at least 31 people, including six al-Qaida members, were killed and 45 other fighters were detained during the clashes.
Iraqi security officials said fighting had decreased since Thursday, but was continuing.