It gets thornier.
Tomorrow, the House Judiciary Committee will vote to authorize subpoenas against the Republican National Committee for emails relevant to the U.S. attorney firings. The RNC, you'll remember, provides email to a number of White House aides, including Karl Rove and his staff.
Chairman John Conyers (D-MI) requested the emails back in April, and yesterday, he got his final answer from the RNC, via the White House: no. In a letter to the RNC's counsel, White House special counsel Emmet Flood wrote that the RNC emails, since they involved White House officials, were covered by the president's assertion of executive privilege last week. The RNC has said that it will abide by the White House's direction. You can see Flood's letter to the RNC here.
The committee's subpoena, if issued later, would add yet another dimension to the fight over documents and testimony and yet another test, if the battle goes to court, of the president's claim of privilege.
Note: House oversight committee Chairman Henry Waxman (D-CA) has also been conducting an investigation of the White House's off-the-record email system.