From the Octahedron to E8

Heres a fun challenge for people confined due to coronavirus.

The E8 lattice is a thing of beauty, taking full advantage of the magic properties of the number 8. The octahedron has 8 sides. Wouldnt it be cool if you could build the E8 lattice from the humble octahedron?

David Harden thinks hes found a way! But I havent carefully checked that it works: it takes some calculations. Can you check it?

He wrote about it here:

David L. Harden, What other lattices are obtainable from this...