For only the fourth time in history, there was consensus–exact correlation–between the Best Picture and the Best Director nominees.
1957: The Bridge on the River Kwai, directed by David Lean
1964: My Fair Lady, directed by George Cukor
1981: Chariots of Fire, directed by Hugh Hudson
2005: Brokeback Mountain, Ang Lee
Notes:
In 1957 and 1964, the Pictures and Directors ended up winning the Oscars.
In 1981, Warren Beatty won Best Director for Reds, but Best icture went to Chariots of Dire.
In 2005, Crash won Best Picture, but Lee won the Best Director