I really don't know that much representation theory or group theory. I was looking at Garrett lisi's presentation, and I was looking at the 3d root system of g2. It struck me that it looked similar to a dodecahedron inscribed in a cube. Now, I do know that the dodecahedron group, and the cube group are dual. Can g2 be broken up into the symmetries of a cube, and a dodecahedron?