Just an illustration of 
[@RichardKlitzing's construction](https://mathoverflow.net/a/378226/6094) in 3D:

     [![CubeDiag][1]][1]

His $n+1 = 4$ layers are:
$$
v_1 \;,\; \{v_2,v_4,v_5\}  \;,\; \{v_3,v_8,v_6\} \;,\; v_7 
$$

  [1]: https://i.sstatic.net/yMdAp.jpg