BRUSSELS  - US Defence Secretary Robert Gates invited Pakistan Thursday to join an investigation into an airstrike that killed 11 Pakistani soldiers, saying he regretted it had created a problem with a key ally. "We think all the procedures were followed but that will be for the investigation to decide, and if we need to make changes we will," Gates told reporters in Brussels on the sidelines of a NATO defence ministers meeting. The US military has until now defended the air strike as a 'legitimate' response to an attack on coalition forces by insurgents who retreated into Pakistan from Afghanistan. The US-led coalition in Afghanistan released a video that it said shows it had targeted insurgents, but Pakistan has expressed outrage over what it said was an 'unprovoked and cowardly' attack on a checkpoint in a volatile tribal area. Gates said he had not personally spoken to the Pakistanis, but the US military had. Admiral Michael Mullen, the Chairman of the Joint Chiefs of Staff, has been in contact with his counterpart in Pakistan. "We agree we need to investigate this incident," Gates said. "We've invited the Afghans and the Pakistanis to be a part of the investigation." "Pakistan is an incredibly important partner for us in this war on terror. Personally, I regret we've had something that has created a problem. We've had an incident that has created a problem between us and the government of Pakistan," he said. Meanwhile, airstrikes by US-led forces in Afghanistan that Pakistan says killed 11 Pakistani soldiers were carried out 'by the book', the top US military chief said in Washington on Thursday. "The details of this (operation) are still not clear (but) every indication I have is that this operation was executed in accordance with procedures, that it was very much by the book," chairman of the joint chiefs Admiral Michael Mullen told reporters. "If there was these deaths of Pakistani military, it is incredibly regrettable," he stressed. So far the United States has not acknowledged having killed Pakistani troops. The US Air Force said on its website that "In Afghanistan, Air Force F-15E Strike Eagles performed shows of force to deter enemy activities in the vicinity of Mita, Orgun-E and Zormat. On-scene joint terminal attack controllers (JTAC) confirmed the missions were successful." "An Air Force B-1B Lancer and F-15Es dropped guided bomb unit-12s, 31s and 38 to destroy anti-coalition members in the open and in buildings in the vicinity of Asadabad. The JTAC reported the missions successful," the report added. Meanwhile, the White House said Thursday that it was still looking into accusations that US-led forces in Afghanistan killed 11 Pakistani soldiers in an air strike, but that it would be 'very saddened' if true. "We're still trying to get to the bottom of what happened. And reports, quite frankly, even from sources within the United States government, are conflicting at this point," said US National Security Adviser Stephen Hadley. "Quite frankly, at this point we have not been able to corroborate that. Should it be true, obviously, we would be very saddened by that loss," Hadley said as US President George W Bush was in Rome on a farewell Europe visit. "You know, Pakistan has been an important ally and will continue - we hope - to be an important ally in the war on terror and we want to help this new democratic government in Pakistan," said Hadley. But "one of the problems is that it is still not exactly clear what happened" in the early hours of Wednesday along Pakistan's remote border with Afghanistan, thought to be a haven for extremists. "What we believe is that there was an operation on the Afghan side of the border by anti-coalition forces that threatened our people, that those forces went back into Pakistan and that we tracked and struck those forces. That's what we believe happened," Hadley told reporters. Hadley said the US ambassador to Pakistan had met with Pakistan's foreign secretary to discuss the incident. Meanwhile, the US-led coalition in Afghanistan deflected accusations that it killed 11 Pakistani soldiers in an airstrike, releasing video footage which it says shows its forces targeting insurgents. The White House appeared to back down on Thursday when it said it was still looking into reports that the coalition had killed Pakistani soldiers but that it would be "very saddened" if they proved accurate. Officials in Washington had earlier said they regretted the "reported loss of Pakistani life" but insisted their forces carried out a "legitimate strike" in the early hours of Wednesday - and had given Pakistan advance warning. On Thursday, the coalition backed those claims with video showing 'precision' strikes on a group of seven rebels who sought refuge in Pakistan after attacking a coalition reconnaissance patrol in eastern Afghanistan. The grainy black and white video footage, taken by an unmanned drone, shows a first strike targeted at men hiding behind a rock who fire two rocket-propelled grenades at coalition troops off-screen. The US-led soldiers were attacked while trying to move to a pick-up point for extraction by helicopter when they were attacked by the "anti-Afghan forces," said a commentary supplied with the video. A huge explosion lights up the area and the militants then seek cover in a deep ravine nearby before three more bombs, one of which is not shown, land on the militants. "It is clear there are no structures or (Pakistani) outposts in the impact area," the voice-over said, adding that all of the militants were killed. But an AFP photographer who reached the scene late Wednesday saw a large crater near a mud-brick Pakistani paramilitary outpost, which was badly damaged. Local tribesmen were scraping pieces of human flesh from nearby trees, while shredded military uniforms and pools of blood littered the scene. Pakistani security officials and residents said the incident began when Afghan troops crossed the porous frontier and tried to occupy a strategic Pakistani post in a long-disputed area of the troubled tribal belt. Residents said word spread in the conservative region and armed tribesmen along with Pakistani Taliban fighters arrived and attacked the Afghan soldiers, who retreated and returned later with coalition forces.