$E_7$ as $D=10$ space-time symmetry --- Origin of the twistor transform
  Massless particle dynamics in $D=10$ Minkowski space is given an $E_7$-covariant formulation, including both space-time and twistor variables. $E_7$ contains the conformal algebra as a subalgebra. Analogous constructions apply to $D=3,4$ and 6. 
Martin Cederwall