A month ago, it was difficult to tell whether President Barack Obama’s heart was in the fight against the Islamic State, the terrorist group that has seized a swath of territory in Iraq and Syria.

His initial statements as the militants roared across the flatlands of northern Iraq focused on the limits of U.S. action. “I will not allow the United States to be dragged into fighting another war,” he said on Aug. 7. Ten days later, he defined the U.S. goal minimally, as seeing that the Islamic State was “contained.”

But by last week, that diffidence was gone. At his news conference after the NATO summit in Wales, Obama said his aim was to “degrade and ultimately destroy” the group, which he called “a significant threat.”

The hardened rhetoric — and, more important, the expanded commitment it promises — came after the Islamic State beheaded two American journalists in Syria and brazenly dared Obama to respond. But even before those grisly killings, administration officials were coming to the conclusion that the group posed a serious threat to U.S. allies in the Middle East and that it could become a source of terrorism against Europe and the United States.

Recognizing the threat is easier than addressing it, though. The strategy, which Obama aides admit they are still “building out,” has a daunting list of moving parts.

It depends on Iraq’s balky politicians to form a new government that can attract support from aggrieved Sunni Muslims.

Then it requires organizing an international coalition against the Sunni-led Islamic State that includes Arab governments in Saudi Arabia, Qatar and Iraq, whose leaders mostly loathe one another.

It will need the Saudis to encourage Iraq’s Sunni Arab tribal leaders to switch their support from the Islamic State to the new Baghdad government, and all the Arab states to stop their citizens from contributing to the militant group.

Meanwhile, in Syria, whose northeastern territory is dominated by the Islamic State, the U.S. and its allies will try to bolster the beleaguered moderate opposition that has been fighting both those militants and the pariah regime of President Bashar Assad.

And the U.S. will need to continue its limited military campaign, both to strike at the authors of terrorist acts against Americans and to stop the Islamists’ advance in northern Iraq (where the U.S. has already launched more than 120 airstrikes and dispatched more than 800 troops).

What won’t be part of the strategy at this point, an administration official told me, are large-scale airstrikes in Syria, because there’s no U.S. ally there that can hold ground once it’s cleared. “You have to have people who can move in to fill that space,” the official said. Otherwise, U.S. airstrikes might merely deliver territory back to the Assad regime.

One unwavering rule underlying Obama’s approach is that he won’t put U.S. combat troops on the ground. Obama has consistently been willing to wage an air war against terrorists (as he has in Pakistan, Yemen and Somalia) but not a ground war. Fighting on the ground will be up to local forces, supported by neighboring governments — which, as U.S. officials point out, are more directly threatened by the Islamic State than we are.

Unlike al-Qaida, the Islamic State uses terrorism as a tactic against its local enemies, not a long-distance international strategy. (That’s one of the reasons it was expelled from al-Qaida, which wants to focus on attacking the United States.) Horrifying as its deeds are, so far they have been aimed only at victims in Iraq, Syria and surrounding countries. They’re happy to attack Americans in their part of the world, but they haven’t shown an inclination to leave the Middle East.

“We have seen no credible information that (the Islamic State) is planning attacks against the homeland,” Obama’s top advisor on terrorism, Lisa Monaco, said last week. What worries her more, she said, is the direct threat the group poses to U.S. allies in the Middle East, as well as the training and experience it is giving recruits from Europe and North America.

Obama’s newly articulated strategy for dealing with that threat won’t be a quick fix. Secretary of State John F. Kerry said it could need three years, which would take us into the tenure of Obama’s successor. The approach won’t satisfy hawks, who want to retaliate any time the group commits a grisly atrocity against civilians. And it won’t mollify doves, who worry that any use of U.S. military power will turn into a slippery slope toward escalation.

Indeed, it may not work at all, since it relies so heavily on cooperation from feuding Arab regimes and local factions. But it’s a sensible beginning — and an important test of whether U.S. military intervention with a light footprint can be made to work.

Doyle McManus is a columnist for the Los Angeles Times.