As the chaos mounts in Baghdad, some Republican politicians have been asking: Who lost Iraq? But that’s a foolish question. Iraq is hobbled by the same sectarian divisions that prevailed long before the Bush administration decided to invade in 2003 — and that were in some ways exacerbated by Bush’s policies during the U.S. military occupation. President Obama didn’t fix the mess of Iraqi politics but he didn’t make it worse, either.

The best thing that could be said about America and Iraq at the chaotic end of 2011 is that the U.S. has moved from military occupier to mediator among feuding Iraqi political factions — a less costly but still frustrating venture.

A symbol of this transition is Gen. Raymond Odierno, the former commander of U.S. forces who returned to Baghdad Thursday to meet with Prime Minister Nouri al-Maliki and his rivals. The meetings were a sign of a continuing U.S. political role, but also its limitations.

Odierno discussed the volatile Iraq situation during an interview aboard his plane Friday as he flew back to the United States. He said his basic message was that all the Iraqi factions should try to work together in the country’s national interest. He stressed to Maliki the importance of working with parliament, sharing power and following the Iraqi constitution.

These recommendations may sound blindingly obvious, but it’s precisely this lack of political compromise that has crippled Iraq since the U.S. overthrew Saddam Hussein. It has turned out that democracy has empowered sectarian politics: Maliki carries all the resentments and suspicions of the Shiite majority toward the Sunnis who formerly ruled the country. The Sunnis, chafing at their minority status, argue that Maliki is adopting Saddam-like dictatorial powers.

Odierno’s schedule in Baghdad illustrated the fragmented political situation — and also his ability to talk with all sides. In addition to meeting Maliki, he saw Osama al-Nujaifi, the Sunni parliament speaker; Rafie al-Issawi, the Sunni finance minister and a leader of Iraqiya; and President Jalal Talabani, a Kurd. Odierno came away hoping that Nujaifi, the parliament speaker, can arrange a meeting soon of the factions.

The Iraqi political feuding neared the flashpoint last week, after the last U.S. troops had left the country. Maliki’s government issued an arrest warrant for Tariq  al-Hashimi, the Sunni vice president, charging that he had plotted to assassinate Shiites in Maliki’s government. To escape capture, Hashimi fled to Kurdistan.

Odierno has had years of experience with this array of politicians. He served as Gen. David Petraeus’ deputy during the troop surge of 2007 that helped bring the country back from near civil war, and then as commander. Odierno is now Army chief of staff and, like Petraeus (who is now CIA director), he retains a strong personal relationship with Maliki that began during the nightmare years of violence.

Odierno’s theme with Maliki was that the U.S. and Iraq have come a long way — and spilled a lot of blood — and that cooperation between the two countries must continue. He warned that Iraq was in danger of becoming a country like Lebanon, where powerful neighbors wage proxy wars and exacerbate sectarian tensions. Unless Maliki pulled the country together, Iraq would be exploited by regional powers Iran, Turkey and Saudi Arabia.

Odierno didn’t appear to make any headway in defusing the charges against Hashimi. Maliki believes that the case — based on a reported confession from a member of Hashimi’s security detail — must be resolved through the Iraqi judicial system. But the Sunnis, who fear that Maliki’s security forces are plotting their own assassination campaign, aren’t likely to defer to the courts.

The effort to work with Maliki has been consistent from the Bush to Obama administrations. It perplexes some (like me) who doubt Maliki’s ability to break from his sectarian roots in the Dawa Party and become a unifying figure. A measure of Washington’s support for Maliki’s government was that before the final troop withdrawal, the U.S. handed over a prisoner named Ali Mussa Daqduq, an Iranian-backed operative who allegedly plotted attacks on U.S. soldiers. The U.S. concluded that under its security framework agreement with Iraq, it had no other choice.

Odierno offered only a brief comment for the record, which conveyed his theme that everyone should calm down: “I’ve learned to not overreact to what goes on politically in Iraq. They seem able to work through their issues. We should watch closely, but we have to respond carefully: It’s about Iraq.”