Tevian Dray, Corinne A. Manogue, Robert A. Wilson
A symplectic representation of $E_7$

Comment.Math.Univ.Carolin. 55,3 (2014) 387-399.

Abstract:We explicitly construct a particular real form of the Lie algebra $\mathfrak e_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak e_7\cong \mathfrak{sp}(6,\mathbb O)$ and, at the group level, $E_7\cong\text{Sp}(6,\mathbb O)$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak e_7$ in terms of rank 3 objects called {\it cubies}.

Keywords: exceptional Lie algebras; octonions; $E_7$
AMS Subject Classification: 17B25 17A35 20G41