The octonionic integers provide a natural description of the $\ee_8$ lattice, as we now show.

Recall that a simple Lie algebra of dimension $d$ and rank $r$ contains a Cartan subalgebra of dimension $r$, consisting of mutually consisting elements. Over $\CC$, the remaining $d-r$ elements can be chosen to be simultaneous eigenvectors of Cartan elements, with real eigenvalues; the resulting $r$-tuples form the root diagram of the Lie algebra, which generates a lattice.

It is intriguing that the root diagram of $\ee_8$, and the corresponding lattice, being 8-dimensional, can be represented using octonions. Since the dimension of $\ee_8$ is 248, and the rank is 8, the root diagram has 240 elements.

One representation of the $\ee_8$ lattice is obtained by first considering the $2^8=256$ elements \begin{equation} \Omega = \left\{\frac12(\pm1\pm i\pm j\pm k\pm k\ell\pm j\ell\pm i\ell\pm\ell)\right\} \end{equation} Clearly, half of these elements have an even number of minus signs; call the collection of such elements $\Omega_+$. We next consider $g_1+g_2$ for pairs of elements $g_1,g_2\in G$, where \begin{equation} G = \{\pm1,\pm i,\pm j,\pm k,\pm k\ell,\pm j\ell,\pm i\ell,\pm\ell\} \end{equation} and where $g_1\ne\pm g_2$; there are ${8\choose2}\times2^2=112$ such elements, which we collectively call $G_0$. Then the root diagram of $\ee_8$ can be given by $\Omega_+\cup G_0$, which generates the odd $\ee_8$ lattice, which we will call $L$, Wilson. The conjugate lattice $R=\bar{L}$ yields another representation of the $\ee_8$ lattice.

A somewhat different representation of the $\ee_8$ lattice is obtained by considering the union of the Hurwitz integers corresponding to all seven quaternionic triples, as well as their complements, namely the 224 Kirmse integers, together with the 16 elements of $G$. This construction yields a third copy of the $\ee_8$ root diagram; the corresponding lattice is the even $\ee_8$ lattice, denoted $B$.

None of the lattices $L$, $R$, $B$ closes under multiplication. However, each of these lattices is closely related to integral octonions! For instance, choose an octonion ($\ell$) to be special, define \begin{equation} a = \frac12 (1+\ell) \end{equation} and denote the $\ell$-integers of § 12.6 by $A$. Then \begin{equation} A = aL = Ra = 2aBa \end{equation} so that \begin{align} L &= 2aA = 2\bar{a}A \\ R &= 2Aa = 2A\bar{a} \\ B &= La = aR = 2aAa \end{align} where we have used the fact that $a=2a^2\bar{a}$, with $2a^2=\ell\in A$. As discussed by Wilson, the Moufang identities now imply that \begin{align} BL &= L \\ RB &= R \\ LR &= 2B \end{align} Thus, the $\ee_8$ lattice has the remarkable property that it can (also) be represented as an integral subalgebra of $\OO$, namely $A$.

Analogous descriptions exist for the other division algebras. The (rescaled) root diagram of $\aa_1=\su(2)$ consists of the two real numbers $\pm1$, which is clearly closed under multiplication. The (rescaled) root diagram of $\aa_2=\su(3)$ is a hexagon, which can be represented as the complex sixth roots of unity—which is again closed under multiplication. Finally, the root diagram of $\dd_4=\so(8)$ is normally given as the ${4\choose2}\times2^2=24$ quaternionic vectors in $G_0$, but a simple rotation (and renormalization) turns this into a copy of the Hurwitz integers—which are the rational quaternionic sixth roots of unity.

Further information about using octonions to describe the $\ee_8$ lattice can be found Wilson.