Physical significance of the cubic invariant in the exceptional Jordan algebra

Ascertain the physical significance of the cubic invariant Tr[J1 ∘ (J2 ∘ J3)] of the exceptional Jordan algebra J3(O) within the proposed E8 × E8 unification framework, and determine how this invariant enters the description of observables or dynamics beyond the quadratic invariant.

Background

In the exceptional Jordan algebra J3(O), both quadratic and cubic forms are preserved by the automorphism group F4. While the quadratic form parallels classical Lie algebra invariants, the cubic form is a distinctive feature of the exceptional case.

The unification program posits that the exceptional Jordan algebra is central to describing fermion generations and potentially deriving fundamental constants, but the specific physical role played by the cubic invariant remains unspecified, highlighting a concrete gap to be addressed.