Part 2 - the integral octonions, 11d supergravity, and cosmological billiards. Also available on the n-Category Café.
Part 3 - the 240 smallest integer octonions, also known as the E_{8} root polytope, and their connection to exceptional objects in mathematics. Also available on the n-Category Café.
Part 7 - Greg Egan's proof that 2 × 2 self-adjoint matrices with integral octonion entries form a copy of the E_{10} lattice. Also available on the n-Category Café.
Part 8 - My proof that 3 × 3 self-adjoint matrices with integral octonion entries form a copy of the K_{27} lattice. Also available on the n-Category Café.