Below we have an image of a rhombic dodecahedron cut according to the A3 hyperplane arrangement (along short diagonals), resulting in 24 cells (= 2 x 12 faces) that correspond to elements of the Weyl group. Another way of thinking about this image is that it is the union of the (closures of) the alcoves neighboring the origin in affine A3.
To build the type A3 Steinberg torus, we identify opposite faces of this subdivided polyhedron.