President Trump’s attempt to fire Lisa Cook, a Federal Reserve governor, signaled a shift in tactics that appeared tailored to that body’s special legal status.

Whether it succeeds will turn on the meaning of a two-word phrase: “for cause.”

In removing the leaders of other independent agencies, Mr. Trump has given no reasons, saying he has the unilateral constitutional authority to control the executive branch, even in the face of congressional efforts to shield officials from political interference. The Supreme Court has lately sided with that view, rejecting statutes that have been on the books for decades requiring presidents to supply a reason for removing officials.

But with the Fed, Mr. Trump has tried something new. This time, he explained in a letter to Ms. Cook on Monday, there was “sufficient cause” to remove her. He accused Ms. Cook of mortgage fraud by designating two different properties as her primary residence in 2021, the year before she was appointed by President Joseph R. Biden Jr.

Ms. Cook responded that “no cause exists under the law” to allow her removal, and her lawyer said she would sue to keep her job.

That lawsuit will raise a series of questions with no definitive answers: What is sufficient cause? Must it be job-related? And who decides?

In Ms. Cook’s case, her actions preceded and so did not arise from her work at the Fed. She has not been charged with a crime, and has not had an opportunity to present her side to a judge or other neutral party.