The tetrahedron based coordinate system from Synergetics generalizes to any number of dimensions easily.
Almost everyone who has written anything about this says vectors from the origin of the coordinate system in the directions of the tetrahedron's vertexes should be added, but they don't add vectors pointed in the directions of the cube's vertexes in the Cartesian coordinate system. The coordinate axes are perpendicular to the planar facets of the cube from the center of volume of the cube in the Cartesian system. The coordinates refer to movements of the planes.
Here are some quotes from Synergetics.
966.20 Tetrahedron as Fourth-Dimension Model: Since the outset of humanity's preoccupation exclusively with the XYZ coordinate system, mathematicians have been accustomed to figuring the area of a triangle as a product of the base and one-half its perpendicular altitude. And the volume of the tetrahedron is arrived at by multiplying the area of the base triangle by one-third of its perpendicular altitude. But the tetrahedron has four uniquely symmetrical enclosing planes, and its dimensions may be arrived at by the use of perpendicular heights above any one of its four possible bases. That's what the fourth-dimension system is: it is produced by the angular and size data arrived at by measuring the four perpendicular distances between the tetrahedral centers of volume and the centers of area of the four faces of the tetrahedron.
962.04 In synergetics there are four axial systems: ABCD. There is a maximum set of four planes nonparallel to one another but omnisymmetrically mutually intercepting. These are the four sets of the unique planes always comprising the isotropic vector matrix. The four planes of the tetrahedron can never be parallel to one another. The synergetics ABCD-four-dimensional and the conventional XYZthree- dimensional systems.
962.03 In the XYZ system, three planes interact at 90 degrees (three dimensions). In synergetics, four planes interact at 60 degrees (four dimensions). re symmetrically intercoordinate. XYZ coordinate systems cannot rationally accommodate and directly articulate angular acceleration; and they can only awkwardly, rectilinearly articulate linear acceleration events.
(Footnote 4: It was a mathematical requirement of XYZ rectilinear coordination that in order to demonstrate four-dimensionality, a fourth perpendicular to a fourth planar facet of the symmetric system must be found--which fourth symmetrical plane of the system is not parallel to one of the already-established three planes of symmetry of the system. The tetrahedron, as synergetics' minimum structural system, has four symmetrically interarrayed planes of symmetry--ergo, has four unique perpendiculars--ergo, has four dimensions.)