An octonionic construction of $E_8$ and the Lie algebra magic square

Abstract

We give a new construction of the Lie algebra of type $E_8$, in terms of $3\times 3$ matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal-Tits magic square of Lie algebras, acting on themselves by commutation.

References in corpus (3)

• Exceptional Lie Algebras, SU(3) and Jordan Pairs Part 2: Zorn-type Representations
• Octions: An $E_8$ description of the Standard Model
• The magic square of Lie groups: the $2\times 2$ case

Cited by in corpus (5)

• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• A New Division Algebra Representation of $E_7$
• A New Division Algebra Representation of $E_6$
• Division Algebras, Triality, and Exceptional Magic