Triality and the Magic Square of Hans Freudenthal

Abstract

We study real triality structures through their intrinsic tensor algebra. Starting from a single triality symbol, we construct the associated Lie algebra of two-triality operators, prove the Jacobi identity, and identify the resulting algebra uniformly with the corresponding entry of the magic square. We then examine the natural invariant bilinear forms and the Clifford-theoretic structures arising from this construction. In low dimension, the triality formalism also recovers classical arithmetic data: in the \(2\times2\times2\) case, the associated binary quadratic forms have a common discriminant and fit naturally into the Bhargava cube picture.