Consider a simple complex Lie group G acting diagonally on a triple flag variety G/P 1 × G/P 2 × G/P 3 , where P i is parabolic subgroup of G. We provide an algorithm for systematically checking when this action has finitely many orbits. We then use this method to give a complete classification for when G is of type F 4 . The E 6 , E 7 , and E 8 cases will be treated in a subsequent paper.