Western military intervention in Libya—which began with air strikes over the weekend—has been described so far as an alliance led by the United States. It's a label the Obama administration appears anxious to hand off, as it faces criticism at home for launching the attack without congressional authorization and seeks to disprove fears that the intervention will become a protracted war without a clear exit plan.

Given the controversy and questions over U.S. involvement—and who's leading what—it’s worth taking a step back to look at exactly how the United States has been involved so far.

What’s happened so far: Coalition strikes have targeted Qaddafi’s troops, air defense, communications and command sites, according to this helpful rundown from Reuters of each round of military action so far and what each coalition country has done.

Who’s leading right now: It’s not clear at the moment, but the New York Times notes that the United States, France and Britain have all been in charge of their own operations up to this point, which the United States—or, to be more specific, the United States Africa Command, or Africom—has been coordinating.

It’s worth noting that the early coordination got off to a rough start, with France pulling the trigger first on Libya, drawing the ire of U.S. and U.K. officials, according to the Financial Times. (French officials maintained that “all air strikes were co-ordinated with our partners.”)

What the United States has said about its involvement: The United States—seeking to allay domestic criticism about its entree into another war—has also stated repeatedly said it will take a step back and transfer leadership over to the broader coalition in a matter of days. Administration officials have said that the U.S. contribution to the mission is targeted and “limited in scope and duration.”

“We will continue to support the efforts to protect the Libyan people, but we will not be leading them,” President Obama said yesterday, after the crash of a U.S. F-15 fighter jet in Libya. Crew members ejected safely. Obama said the transition would occur in “days, not weeks.”

“We came in up front fairly heavily, fairly substantially, at considerable risk to our military personnel,” he said. “When this transition takes place, it is not going to be our planes that are maintaining the no fly zone. It is not going to be our ships that are necessarily enforcing the arms embargo. That’s precisely what the other coalition partners are going to do.”

Why there’s been a hold-up on the transfer of leadership: The squabble has been about who will lead that coalition. Britain, Italy and the United States favored using NATO to coordinate. France did not. The Times reports that Turkey, which has kept diplomatic channels open with Libya, has also been reluctant—it is the only Muslim member of NATO and opposed military action against Libya.

The United States, however, has been working to get Turkey on board, with President Obama calling Turkey’s prime minister yesterday. According to the summary of the call, the two “agreed that this will require a broad-based international effort, including Arab states, to implement and enforce the UN resolutions, based on national contributions and enabled by NATO’s unique multinational command and control capabilities.”

The Financial Times reported on Tuesday that members of the alliance were nearing an agreement as to a new command structure using NATO’s planning capabilities but including other non-NATO countries via a steering committee. (Foreign Policy laid out a few other options as well.) The Post reported late Tuesday that the tentative agreement has been reached.

Why regional support is a concern: While nations in the Arab League and Gulf Cooperation Council had been calling for the United Nations to consider a no-fly zone and protection of civilians in Libya, keeping some of these nations on board—key to the mission’s legitimacy—has been a diplomatic task unto itself.

Regional support seemed to waver recently when the Arab League’s secretary general criticized the action against Libya as exceeding the UN mandate.

The United States made a number of frantic calls, the Guardian reported, and yesterday the Arab League clarified its support: “We are committed to UN security council resolution 1973. We have no objection to this decision, particularly as it does not call for an invasion of Libyan territory.”

“There was a broad set of, really, requests and demands from the region for action,” National Security Adviser Tom Donilon told reporters on Monday. “The Gulf Cooperation Council and the Arab League came forward and asked for the United Nations Security Council to step up and address this issue.”

The Journal reports that while the Gulf Cooperation Council, a group of six Persian Gulf states, has stayed steadfast in its support, it hasn’t offered much in the way of military aid:

Meanwhile, the Gulf Cooperation Council, a group of six Arab Persian Gulf states, has voiced unwavering political support for the strikes. The bloc was the first among Arab states to urge the U.N. to consider a no-fly zone at a meeting early in the month. But apart from Qatar, which is sending jets, GCC members have stopped short of sending military aid.

The Atlantic has a few lists on where each Middle East country stands on the Libya action and what the various countries have contributed or pledged to contribute. (Note that the Atlantic, citing AFP, said that the United Arab Emirates is sending aircraft—but that's now changed because the UAE is unhappy about the U.S. and European stance on Bahrain.)

What’s been promised by the coalition: The coalition has stated that in accordance with the United Nations resolution, no ground troops will be sent in. Both the United States and U.K. have said that they are not going after Qaddafi.

“The current focus is on the protection of civilians in Libya,” said Donilon. “That’s what the military tools are being used right now to do.”

Four days of airstrikes, however, have so far seemed ineffective in preventing Qaddafi's attacks, according to the Post. Libyan forces continued attacking civilians in the city of Misurata yesterday, though the Associated Press reports that the strikes eventually forced Qaddafi's tanks to retreat.

As we’ve noted, it had earlier been reported there had been a “growing consensus” within the White House that a no-fly zone over Libya would no longer make a difference. That was before the United States voted in favor of military intervention.

Pentagon officials over the weekend said that the early strikes had put the no-fly zone “effectively in place,” though Qaddafi “clearly still has the capability to attack his own people.”