A new division algebra representation of E6 from E8

Abstract

We construct the well-known decomposition of the Lie algebra e8 into representations of e6⊕su(3) using explicit matrix representations over pairs of division algebras. The minimal representation of e6, namely the Albert algebra, is thus realized explicitly within e8, with the action given by the matrix commutator in e8, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in e8, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group E6 ⊂ E8.