Exceptional Twistor Theory II: Generalization to $E_7$ and $E_8$

We extend the octonionic twistor program from $E_6$ to the larger exceptional Lie groups $E_7$ and $E_8$. For $E_7$, the twistor space is the 56-dimensional Freudenthal triple system H = T ⊕ T ∗ , equipped with a quartic invariant that yields a fundamental 4-vertex intertwiner. For E8 , we identify an 84-dimensional candidate twistor space carrying the Cartan cubicinvariant, which provides a 3-vertex structure. Both extensions preserve the key features of the E6 theory: the Penrose transform generalizes to integration over appropriate cycles, and non-associativity is handled uniformly via slice-regular functions. This unified framework, organized by the Freudenthal–Tits magic square, offers a systematic approach to physicswith exceptional symmetry.