Triality and the Magic Square of Hans Freudenthal

TL;DR

This paper explores real triality structures via tensor algebra, linking them to Lie algebras, the magic square, and classical arithmetic data, revealing new algebraic and geometric insights.

Contribution

It constructs Lie algebras from triality symbols, proves their properties, and connects triality to classical forms and the Bhargava cube framework.

Findings

Constructed Lie algebras from triality symbols and proved Jacobi identity.

Identified algebraic structures with entries of the magic square.

Connected triality formalism to classical quadratic forms and Bhargava's cube.

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.