We describe two constructions of a certain ${\mathbb{Z}}_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension $56$ and skew-dimension $1$) over an algebraically closed field of characteristic different from $2$. The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types $E_6$, $E_7$ and $E_8$.

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.