WASHINGTON  On the plane, no longer Air Force One but now Special Air Mission 28000, they talked about the speech. George W. Bush, the former president, was heading home to Texas with his inner circle, having just left the Capitol, where his successor first thanked him for his service and then proceeded to discredit it.

The Bush team had worked assiduously to make the transition smooth for the incoming President Obama and stayed out of the way as he used the postelection period to take leadership of the economy even before being sworn in. And now, as far as some of them were concerned, the new president had used his inaugural lectern to give the back of the hand to a predecessor who had been nothing but gracious to him.

“There were a few sharp elbows that really rankled and I felt were not as magnanimous as the occasion called for,” Karen Hughes, a longtime Bush confidante, said in an interview. “He really missed an opportunity to be as big as the occasion was and, frankly, as gracious as President Bush was as he left office.”

Dan Bartlett, another top adviser, used similar language. “It was a missed opportunity to bring some of the president’s loyal supporters into the fold,” he said. Marc A. Thiessen, the chief White House speechwriter until this week, added: “It was an ungracious inaugural. It was pretty clear he was taking shots.”