A new division algebra representation of E7 from E8

Abstract

We decompose the Lie algebra e8(−24) into representations of e7(−25)⊕sl(2,R) using our recent description of e8 in terms of (generalized) 3 × 3 matrices over pairs of division algebras. Freudenthal’s description of both e7 and its minimal representation are therefore realized explicitly within e8, with the action given by the (generalized) matrix commutator in e8, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in e8.