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Keywords:
exceptional Lie algebras; octonions; $E_7$
exceptional Lie algebras; octonions; $E_7$
Summary:
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.
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