Higgsy Codes III

On Cullinane's page on the miracle octad generator we find a beautiful note by Robert Wilson, the Wilson of the new Leech lattice, for R. T. Curtis' 60th birthday in 2007. The note looks at Lie algebra root systems in terms of integral quaternions and octonions. He defines an order $24$ group

$\omega \equiv \frac{1}{2} (-1 + i + j + k)$

with elements $\{ \pm 1, \pm i, \pm j, \pm k \}$ and $\{ 1, \omega, \overline{\omega} \}$, and the neat relation $i^{\omega} = j$. This gives a root system for $D_4$ and is extended using norm $2$ elements to a root system for $F_4$. Of course, with the octonions, we get $E_8$. Morally, the octonions are responsible for all exceptional Lie algebras. The exceptional algebras $G_2$, $F_4$ and $E_8$ have orders $14$, $52$ and $248$, with respective prime factors

$7 = 1 + 2 + 2^2$
$13 = 1 + 3 + 3^2$
$31 = 1 + 5 + 5^2$.

Wilson spots the combinatorics of finite projective planes, but then notes we can also write the triality

$14 = 2 (2^3 - 1) = 16 - 2$
$52 = 2 (3^3 - 1) = 54 - 2$
$248 = 2 (5^3 - 1) = 250 - 2$

where, as kneemo keeps saying, these numbers appear in qutrit symmetry groups. Recall that $14$ also counts the vertices of the three dimensional associahedron, and $16$ the extension to crossing partitions, while $54$ is the dimension of the bioctonion algebra. Speaking of trialities, the largest Heegner number is also written as

$163 = 1 + 1 + 1 + 2^3 + 3^3 + 5^3 = 54 + 54 + 54 + 1$

where $2^3$ counts vertices on a parity cube, $3^3$ paths for the tetractys, and $5^3$ paths for $5$-valued states.