There is an unwritten tradition in Washington that cabinet secretaries all submit pro forma letters of resignation at the end of a first term for a re-elected president, but apparently it doesn’t really work that way.
The last two-term president was George W. Bush and according to an official in the West Wing at the time, such letters were requested, but some secretaries had to be prodded to comply. Some notable figures departed the administration including Secretary of State Colin Powell.
Bill Clinton was the last Democratic two-term president and one of his cabinet secretaries recalls being called in for a personal meeting in the Oval Office. She had intended to resign and return to academia. The president refused to accept the resignation and she remained for the entire eight years.
In recent memory, only one cabinet official remained in office from one president to the next: Former Defense Secretary Robert Gates was managing two wars at the end of the Bush administration and he remained at the Pentagon at President Obama’s request.
The constitutional powers of a president expire at noon on January 20th every four years but cabinet secretaries serve at the pleasure of the president, which means those resignation letters are considered an unwritten requirement at the end of any president’s second term.