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Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group $E_6$, and of its subgroups. We are therefore led to a description of $E_6$ in terms of $3\times 3$ octonionic matrices, generalizing previous results in the $2\times 2$ case. Our treatment naturally includes a description of several important subgroups of $E_6$, notably $G_2$, $F_4$, and (the double cover of) $SO(9,1)$. An interpretation of the actions...

We explicitly construct a particular real form of the Lie algebra ${𝔢}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications ${𝔢}_7\cong \mathrm{𝔰𝔭}(6,𝕆)$ and, at the group level, $E_7\cong \text{Sp}(6,𝕆)$. Along the way, we provide a geometric description of the minimal representation of ${𝔢}_7$ in terms of rank 3 objects called cubies.