If you want to know when American troops can draw down from Iraq, here’s the nitty-gritty:

The Bush administration has lost control over that decision. America’s fate in Iraq depends not on the skill of U.S. troops, but on the abilities of feuding Iraqi political factions to form a government that can prevent civil war.

U.S. generals know the Iraqi security forces they are training won’t hold together unless Iraqi politicians do. If Iraqis divide by sect, their security forces will do the same. Iraqi forces didn’t stop the torching of mosques in sectarian violence last month.

So the president’s mantra that as “Iraqis stand up, we’ll stand down” is a pipe dream if Sunni, Kurd and Shiite politicians can’t sit together. Yet nearly three months after elections, Iraq still has no new government.

The situation isn’t hopeless. But much depends on the finale of the dangerous political drama playing out in Baghdad, whose key actors include the U.S. ambassador to Baghdad, Zalmay Khalilzad, the radical Shiite cleric Muqtada al-Sadr, and the supreme Shiite cleric Ayatollah Ali al-Sistani. These three will shape the script in coming weeks.

The drama revolves around who will become Iraq’s next prime minister. The current prime minister, Ibrahim al-Jaafari, has been tapped to keep his job by the Shiite bloc that won the most seats in January elections. But al-Jaafari showed little competence in his first term and won the nod within his own bloc by only one vote.

Al-Jaafari’s most powerful backer is the radical al-Sadr, who commands a militia, known as the Mahdi army, that has killed many Americans and Iraqis. If the weak al-Jaafari takes office, beware al-Sadr’s role.

In the explosion of sectarian violence that followed the recent bombing of a Shiite shrine, the Mahdi army reportedly attacked many Sunni mosques. But the clever al-Sadr, who was outside Iraq during the fighting, swept back to Baghdad and called for Shiite-Sunni unity.

Al-Sadr is trying to position himself as a mediator, meeting radical Sunni clerics and Sunni Arab leaders outside Iraq. But his real stripes are revealed by his militia. I’ve met many Shiites in Baghdad who relate how his followers keep control by murder and extortion.

The black-turbaned cleric espouses an activist brand of Shiite Islam that was promulgated by his father, a populist ayatollah murdered by Saddam Hussein. Young al-Sadr seeks a prominent governmental role for clerics, like the role Shiite mullahs play in Iran. Such a role is anathema to Iraq’s Shiite clerical establishment, which espouses a quietist theology and privately detests al-Sadr. With al-Jaafari in power, al-Sadr’s thugs would play havoc.

Many Shiites fear al-Sadr and would like al-Jaafari to step aside for a more competent choice, the Shiite Adel Abdul-Mahdi. But according to his Shiite colleagues, al-Jaafari believes God has chosen him for the job (sound familiar?) and refuses to step down.

Al-Jaafari has angered the Kurds by opposing their claims on the city of Kirkuk. Many Sunnis mistrust al-Sadr because he failed to prevent his men from attacking their mosques. Sunnis also know al-Jaafari didn’t prevent the infiltration of the Interior Ministry’s security forces by Shiite militiamen who murder Sunni civilians.

So Sunni, Kurdish and secular parties, who together control about half the parliamentary seats, have united to block al-Jaafari’s nomination. But because the constitution requires a two-thirds parliamentary vote to form a government, the whole process is frozen. Unless the stalemate is broken, Iraq won’t have a government and the U.S. exit strategy will collapse.

What is to be done?

Here’s where Ayatollah al-Sistani and ambassador Khalilzad play key roles. The ayatollah needs to inform Shiite parliamentarians that he won’t object if some join the Kurds and Sunnis in opposing al-Jaafari. That could produce the necessary two-thirds margin. Until now, the ayatollah has opposed any split in the Shiite bloc lest it dilute the power the Shiite majority struggled for decades to achieve.

Once the numbers shift, Khalilzad should undertake the initiative he proposed in a Time interview: to persuade leaders of all Iraqi political factions to gather, away from Baghdad, and stay put until they work out an agenda for national unity.

I watched Khalilzad work similar magic in London in December 2002, when he corralled members of the Iraqi opposition and kept them talking in a 14th-floor hotel suite until they reached accord in the wee hours.

Such mediation is crucial at this moment.

We need a new actor for prime minister, an end to al-Sadr’s star role, and a new script for Iraqi unity. Only then can Iraq move forward and U.S. troops draw down.