IN LAUNCHING two previous wars in Iraq, the United States assembled formidable coalitions of dozens of countries. Saudi Arabia, Egypt, Syria and Oman were among the Arab states that deployed substantial ground forces during the 1991 Persian Gulf War. Though derided by some as a “unilateral” U.S. action, the 2003 invasion and subsequent occupation of Iraq was supported by troops from 39 countries, nine of which deployed more than 1,000 soldiers.

By those standards, the results thus far of the Obama administration’s efforts to marshal an alliance to fight the self-described Islamic State look meager. In Paris on Monday, two dozen governments pledged to help fight the extremists “by any means necessary, including military assistance.” But only a handful — not yet including Britain — have so far agreed to participate in air combat missions in Iraq, and none has yet signed on to support prospective U.S. air strikes in Syria. Nor is any sending combat troops.

The attenuated support reflects in part the complicated politics of the fight against the Islamic State, which controls a New England-size swath of territory across Iraq and Syria and commands tens of thousands of militants. Neighbors such as Turkey and Jordan are reluctant to openly join the fight, for fear of becoming targets of the terrorists. Sunni rulers are loath to fight on the same side as the Syrian regime of Bashar al-Assad or Iran — which, for its part, loudly declared Monday that it had rejected a U.S. cooperation proposal.

In large part, however, the restraint has been fostered by President Obama himself. As The Post’s Rajiv Chandrasekaran reported, Mr. Obama rejected the recommendation of his top military commanders that U.S. Special Operations forces be deployed to assist Iraqi army units in fighting the rebels, and Secretary of State John F. Kerry said the administration has turned aside troop offers by other nations. “There are some who have offered to do so, but we are not looking for that, at this moment anyway,” he told CBS News’s Bob Schieffer.

Mr. Kerry said Monday that defeating the Islamic State will depend in part on non-combat initiatives such as discrediting its ideology, stopping the flow of jihadist volunteers and providing political and material support to the new Iraq government. That’s certainly true, and Saudi Arabia’s commitment to help train thousands of fighters for the Free Syrian Army is important.

Still, it’s not clear that the administration’s strategy of defeating the Islamic State solely with Iraqi and rebel Syrian forces is workable. The Iraqi army will require extensive reorganization and retraining, backed by major political reforms in Baghdad, before it will be able to field units capable of recapturing cities such as Fallujah and Mosul. And it looks strong compared with the Free Syrian Army, which until recently appeared in danger of being crushed between the Assad regime and the Islamic State.

Mr. Obama is right to seek the empowerment of the Iraqi and Syrian forces and to fashion a broad agenda for a regional alliance. But in the end the Islamic State will have to be defeated on the battlefield. In that respect, the alliance the administration is constructing looks underpowered.