The White House clarified that Mr. Musk did not work for the newly created Department of Government Effectiveness, although the name was seen as almost synonymous with it. Joshua Fisher, director of the White House Executive Office, said Mr. Musk was a senior adviser to the president and had no higher authority than other senior White House staff. Mr. Fisher said late Monday that Mr. Musk's role was separate from the Department of Government Effectiveness, which he supports.