There exist only 13 manners to split a cube in tetrahedras built exclusively with the 8 corners of the cube without counting simple rotations und reflections of the splittings.
Using the following notation
the splitting can be described by a list of 4 corner numbers for each tetrahedra.
There are 12 splittings in 6 tetrahedra and one in 5 tetrahedra.
The number 13 is
and number 1 is
The following is a stereoscopic view of an exploded splitting (No 3)