You need to chop of four corners with an equilateral triangle as the base with sides of diagonals
Four of them
That leaves a tetrahedron in the middle
We need to cut it in four equal parts with each equilateral triangular base (four of them) and apex meeting at the center of the cube
The four earlier cut pieces (cut for the sake of explanation) are merged with equal area bases of the four pieces of the tetrahedron