A Kind of Magic

It is a wondrous result that one can construct the Freudenthal Magic Square from the normed division algebras.  Using tensored division algebras to generate the Magic Square has also proven to be useful in the double copy procedure for Yang-Mills theories and gravity, resulting in the Magic Pyramid.  However, Landsberg and Manivel found midway between e7 and e8 there should be a non-reductive Lie algebra e7(1/2), which is related to the 'sextonions', a six-dimensional algebra midway between the quaternions and octonions.  This implies a generalization of the Freudenthal Magic Square and Magic Pyramid.  Marrani and Borsten took this analysis to its logical completion in A Kind of Magic, by filling in the "missing gaps" with the 3-dimensional "ternions" and the sextonions, and even studying their U-duality counterparts.  On the M-theory side, one can then ask, what type of compactification leads to such non-reductive Lie algebraic symmetries?  Or even better, are these symmetries hinting at something larger, beyond 11-dimensions?