Agency officials say that because the institute is a congressionally chartered nonprofit that is not part of the executive branch, Mr. Trump and Mr. Musk do not have the authority to gut its operations.

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George Moose, who was fired as the institute’s acting president last week but is challenging his dismissal, accused Mr. Musk’s team of breaking in. “Our statute is very clear about the status of this building and this institute,” he told reporters. “So what has happened here today is an illegal takeover by elements of the executive branch of a private nonprofit corporation.”