The Magic Star

In arXiv:1112.1258 [math-ph], P. Truini gave an elegant construction of the exceptional algebras, via a star-like projection under an A2 (su(3)).  For the F4, E6, E7, E8 cases, the 6 star vertices contain root vectors that transform as a Jordan algebra of degree three (containing 3x3 hermitian elements over the division algebras A=R, C, H, O).  In the center is the reduced structure group of the Jordan algebra.

In the maximal case, the Jordan algebra is the exceptional one, first mentioned in the famous quantum mechanical classification paper by Jordan, Wigner and von Neumann (1934).  This "E8 Magic Star" construction makes the symmetries of extremal black holes in D=4 N=8 SUGRA manifest, as one can take vertical slices and notice the 5-grading 1+56+(E7+1)+56+1 appears naturally.  Another application of the star would be to identify 27-dimensional Jordan algebraic star vertex as the matrix degrees of freedom in a "bosonic M-theory" Chern-Simons theory and notice the reduced structure group E6 is the full invariance group of its cubic form.  P. Truini and A. Marrani also proposed an emergent model based on this star projection.  (Note: there is also a spectacular way to extend this star beyond E8, coming in a forth-coming paper with the author and P. Truini and A. Marrani. Stay tuned!)

In Garrett Lisi's elementary particle explorer, the E8 Magic Star is visualized as:

One can view it by moving the V direction under the E6 Coxeter view mode with these settings: