In 2D, we have the parallelogons:

- The square leads to the square tiling {4,4}. The 2 axes can be used directly for the Cartesian 2D coordinate system.
- The hexagon leads to the hexagonal tiling {6,3}. Any 2 of the 3 axes can be used as a basis for a coordinate system.

In 3D, we have the parallelohedra:

- The cube {4,3} leads to the cubic honeycomb {4,3,4}. The 3 axes can be used directly for the Cartesian 3D coordinate system.
- The hexagonal prism (6,4,4) leads to the hexagonal prismatic honeycomb. The hexagon axis (i.e. the one orthogonal to the hexagon faces) can be used with any 2 of the 3 square axes for a coordinate system.
- The truncated octahedron (6,6,4) leads to the bitruncated cubic honeycomb 2t{4,3,4}. Any set of 3 hexagon axes can be used for a coordinate system, as well as any set of 1 hexagon axis and 2 square axes. There's also the option to use 2 hexagon axes, then pick a square that is not directly between 2 hexagons from those axes, and use the axis orthogonal to this square, too.
- The rhombic dodecahedron leads to the rhombic dodecahedral honeycomb. The 3 axes associated with the faces around an order-3 vertex can be used for a coordinate system. Another option is to pick 2 axes on a hexagonal equator, and add the axis from a face pair that is not on this equator.
- The elongated dodecahedron basically leads to the same coordinate system as the one from the rhombic dodecahedra. The only difference is that one axis is stretched (assuming maximum symmetry), and there's a difference which cells share an edge or a vertex.

Now, 4D is a lot more complicated, and there is a total of 52 parallelotopes. So I was wondering how much this number can be reduced by removing parallelotopes that lead to mostly redundant coordinate systems (like the grids for the elongated dodecahedron and the rhombic dodecahedron in 3D).

For a start, we can take the parallelohedra, expand them along a 4th axis that is orthogonal to the original 3D space. If we do this with the cube, hexagonal prism, truncated octahedron, and rhombic dodecahedron, we get 4 parallelotopes that can be used for unique 4D coordinate systems:

- Expanding the cube gives us the tesseract, which can be used for a 4D Cartesian coordinate system.
- Expanding the hexagonal prism results in a parallelotope with 4 hexagonal prisms and 6 cubes.
- Expanding the truncated octahedron results in a parallelotope with 2 truncated octahedra, 8 hexagonal prisms, and 6 squares.
- Expanding the rhombic dodecahedron results in a parallelotope with 2 rhombic dodecahedra and 12 parallelepipeds.

We could also do the same with the elongated dodecahedron, but remove the resulting parallelotope due to a (mostly) redundant grid.

To get another (parallelotope-based?) coordinate system, it should be possible to take the coordinates from two hex grids, and combine both for the coordinates of a 4D grid (think of each hex grid as representing a plane, and both planes only intersect in the origin of the coordinate system). I wouldn't be surprised if the Voronoi diagram from all points with integer coordinates in this coordinate system would be a parallelotope tiling, but my intuition about 4-dimensional space is pretty bad.

Is there a simple way to get a complete list of 4D parallelotopes with unique coordinate systems, i.e. with redundant parallelotopes removed?