Geodesic
A symplectic representation of $E_7$
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 387-399

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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 cubies.
Classification : 17A35, 17B25, 20G41
Keywords: exceptional Lie algebras; octonions; $E_7$ 
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Dray, Tevian; Manogue, Corinne A.; Wilson, Robert A. A symplectic representation of $E_7$. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 3, pp. 387-399. http://geodesic.mathdoc.fr/item/CMUC_2014__55_3_a7/