This time I’d like to summarize some work I did in the comments last time, egged on by a mysterious entity who goes by the name of ‘Metatron’.

As you probably know, there’s an archangel named Metatron who appears in apocryphal Old Testament texts such as the Second Book of Enoch. These texts rank Metatron second only to YHWH himself. I don’t think the Metatron posting comments here is the same guy. However, it’s a good name for someone interested in lattices and geometry, since there’s a variant of the Cabbalistic Tree of Life called **Metatron’s Cube**, which looks like this:

This design includes within it the $<semantics>{\mathrm{G}}_{2}<annotation\; encoding="application/x-tex">\backslash mathrm\{G\}\_2</annotation></semantics>$ root system, a 2d projection of a stellated octahedron, and a perspective drawing of a hypercube.

Anyway, there are lattices in 26 and 27 dimensions that play rather tantalizing and mysterious roles in bosonic string theory. Metatron challenged me to find octonionic descriptions of them. I did.

Given a lattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ in $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$-dimensional Euclidean space, there’s a way to build a lattice $<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$ in $<semantics>(n+2)<annotation\; encoding="application/x-tex">(n+2)</annotation></semantics>$-dimensional Minkowski spacetime. This is called the ‘over-extended’ version of $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$.

If we start with the lattice $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$ in 8 dimensions, this process gives a lattice called $<semantics>{\mathrm{E}}_{10}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{10\}</annotation></semantics>$, which plays an interesting but mysterious role in superstring theory. This shouldn’t come as a complete shock, since superstring theory lives in 10 dimensions, and it can be nicely formulated using octonions, as can the lattice $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$.

If we start with the lattice called $<semantics>{\mathrm{D}}_{24}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}</annotation></semantics>$, this over-extension process gives a lattice $<semantics>{\mathrm{D}}_{24}^{++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{++\}</annotation></semantics>$.
This describes the ‘cosmological billiards’ for the 3d compactification of the theory of gravity arising from bosonic string theory. Again, this shouldn’t come as a complete shock, since bosonic string theory lives in 26 dimensions.

Last time I gave a nice description of $<semantics>{\mathrm{E}}_{10}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{10\}</annotation></semantics>$: it consists of $<semantics>2\times 2<annotation\; encoding="application/x-tex">2\; \backslash times\; 2</annotation></semantics>$ self-adjoint matrices with integral octonions as entries.

It would be nice to get a similar description of $<semantics>{\mathrm{D}}_{24}^{++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{++\}</annotation></semantics>$. Indeed, one exists! But to find it, it’s actually easier to go up to 27 dimensions, because the space of $<semantics>3\times 3<annotation\; encoding="application/x-tex">3\; \backslash times\; 3</annotation></semantics>$ self-adjoint matrices with octonion entries is 27-dimensional. And indeed, there’s a 27-dimensional lattice waiting to be described with octonions.

You see, for any lattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ in $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$-dimensional Euclidean space, there’s also a way to build a lattice $<semantics>{L}^{+++}<annotation\; encoding="application/x-tex">L^\{+++\}</annotation></semantics>$ in $<semantics>(n+3)<annotation\; encoding="application/x-tex">(n+3)</annotation></semantics>$-dimensional Minkowski spacetime, called the ‘very extended’ version of $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$.

If we do this to $<semantics>L={\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">L\; =\; \backslash mathrm\{E\}\_8</annotation></semantics>$ we get an 11-dimensional lattice called $<semantics>{\mathrm{E}}_{11}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{11\}</annotation></semantics>$, which has mysterious connections to M-theory. But if we do it to $<semantics>{\mathrm{D}}_{24}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}</annotation></semantics>$ we get a 27-dimensional lattice sometimes called $<semantics>{\mathrm{K}}_{27}<annotation\; encoding="application/x-tex">\backslash mathrm\{K\}\_\{27\}</annotation></semantics>$. You can read about both these lattices here:

I’ll prove that both $<semantics>{\mathrm{E}}_{11}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{11\}</annotation></semantics>$ and $<semantics>{\mathrm{K}}_{27}<annotation\; encoding="application/x-tex">\backslash mathrm\{K\}\_\{27\}</annotation></semantics>$ have nice descriptions in terms of integral octonions. To do this, I’ll use the explanation of over-extended and very extended lattices given here:

These constructions use a 2-dimensional lattice called $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$. Let’s get to know this lattice. It’s very simple.

### A 2-dimensional Lorentzian lattice

Up to isometry, there’s a unique even unimodular lattice in Minkowski spacetime whenever its dimension is 2 more than a multiple of 8. The simplest of these is $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$: it’s the unique even unimodular lattice in 2-dimensional Minkowski spacetime.

There are various ways to coordinatize $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$. The easiest, I think, is to start with $<semantics>{\mathbb{R}}^{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^2</annotation></semantics>$ and give it the metric $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ with

$$<semantics>g(x,x)=-2uv<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; -2\; u\; v\; </annotation></semantics>$$

when $<semantics>x=(u,v)<annotation\; encoding="application/x-tex">x\; =\; (u,v)</annotation></semantics>$. Then, sitting in $<semantics>{\mathbb{R}}^{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^2</annotation></semantics>$, the lattice $<semantics>{\mathbb{Z}}^{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}^2</annotation></semantics>$ is even and unimodular. So, it’s a copy of $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$.

Let’s get to know it a bit. The coordinates $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$ and $<semantics>v<annotation\; encoding="application/x-tex">v</annotation></semantics>$ are called **lightcone coordinates**, since the $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$ and $<semantics>v<annotation\; encoding="application/x-tex">v</annotation></semantics>$ axes form the lightcone in 2d Minkowski spacetime. In other words, the vectors

$$<semantics>\ell =(1,0),\phantom{\rule{1em}{0ex}}\ell \prime =(0,1)<annotation\; encoding="application/x-tex">\; \backslash ell\; =\; (1,0),\; \backslash quad\; \backslash ell\text{\'}\; =\; (0,1)\; </annotation></semantics>$$

are **lightlike**, meaning

$$<semantics>g(\ell ,\ell )=0,\phantom{\rule{1em}{0ex}}g(\ell \prime ,\ell \prime )=0<annotation\; encoding="application/x-tex">\; g(\backslash ell,\backslash ell)\; =\; 0\; ,\; \backslash quad\; g(\backslash ell\text{\'},\; \backslash ell\text{\'})\; =\; 0\; </annotation></semantics>$$

Their sum is a timelike vector

$$<semantics>\tau =\ell +\ell \prime =(1,1)<annotation\; encoding="application/x-tex">\; \backslash tau\; =\; \backslash ell\; +\; \backslash ell\text{\'}\; =\; (1,1)</annotation></semantics>$$

since the inner product of $<semantics>\tau <annotation\; encoding="application/x-tex">\backslash tau</annotation></semantics>$ with itself is negative; in fact

$$<semantics>g(\tau ,\tau )=-2<annotation\; encoding="application/x-tex">\; g(\backslash tau,\backslash tau)\; =\; -2\; </annotation></semantics>$$

Their difference is a spacelike vector

$$<semantics>\sigma =\ell -\ell \prime =(1,-1)<annotation\; encoding="application/x-tex">\; \backslash sigma\; =\; \backslash ell\; -\; \backslash ell\text{\'}\; =\; (1,-1)\; </annotation></semantics>$$

since the inner product of $<semantics>\sigma <annotation\; encoding="application/x-tex">\backslash sigma</annotation></semantics>$ with itself is positive; in fact

$$<semantics>g(\sigma ,\sigma )=2<annotation\; encoding="application/x-tex">\; g(\backslash sigma,\backslash sigma)\; =\; 2\; </annotation></semantics>$$

Since the vectors $<semantics>\tau <annotation\; encoding="application/x-tex">\backslash tau</annotation></semantics>$ and $<semantics>\sigma <annotation\; encoding="application/x-tex">\backslash sigma</annotation></semantics>$ are orthogonal and have length $<semantics>\sqrt{2}<annotation\; encoding="application/x-tex">\backslash sqrt\{2\}</annotation></semantics>$ in the metric $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$, we get a square of area $<semantics>2<annotation\; encoding="application/x-tex">2</annotation></semantics>$ with corners

$$<semantics>0,\tau ,\sigma ,\tau +\sigma <annotation\; encoding="application/x-tex">\; 0,\; \backslash tau,\; \backslash sigma,\; \backslash tau\; +\; \backslash sigma\; </annotation></semantics>$$

that is,

$$<semantics>(0,0),\phantom{\rule{thickmathspace}{0ex}}(1,1),\phantom{\rule{thickmathspace}{0ex}}(1,-1),\phantom{\rule{thickmathspace}{0ex}}(2,0)<annotation\; encoding="application/x-tex">\; (0,0),\backslash ;\; (1,1),\backslash ;\; (1,-1),\; \backslash ;(2,0)\; </annotation></semantics>$$

If you draw a picture, you can see by dissection that this square has twice the area of the unit cell

$$<semantics>(0,0),\phantom{\rule{thickmathspace}{0ex}}(1,0),\phantom{\rule{thickmathspace}{0ex}}(0,1),\phantom{\rule{thickmathspace}{0ex}}(1,1)<annotation\; encoding="application/x-tex">\; (0,0),\backslash ;\; (1,0),\; \backslash ;\; (0,1)\; ,\; \backslash ;\; (1,1)\; </annotation></semantics>$$

So, the unit cell has area 1, and the lattice is **unimodular** as claimed. Furthermore, every vector in the lattice has even inner product with itself, so this lattice is **even**.

### Over-extended lattices

Given a lattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ in Euclidean $<semantics>{\mathbb{R}}^{n}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^n</annotation></semantics>$,

$$<semantics>{L}^{++}=L\oplus \mathrm{H}<annotation\; encoding="application/x-tex">L^\{++\}\; =\; L\; \backslash oplus\; \backslash mathrm\{H\}\; </annotation></semantics>$$

is a lattice in $<semantics>(n+2)<annotation\; encoding="application/x-tex">(n+2)</annotation></semantics>$-dimensional Minkowski spacetime, also known as $<semantics>{\mathbb{R}}^{n+1,1}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^\{n+1,1\}</annotation></semantics>$. This lattice $<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$ is called the **over-extension** of $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$.

A direct sum of even lattices is even. A direct sum of unimodular lattices is unimodular. Thus if $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is even and unimodular, so is $<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$.

All this is obvious. But here are some deeper facts about even unimodular lattices. First, they only exist in $<semantics>{\mathbb{R}}^{n}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^n</annotation></semantics>$ when $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ is a multiple of 8. Second, they only exist in $<semantics>{\mathbb{R}}^{n+1,1}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^\{n+1,1\}</annotation></semantics>$ when $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ is a multiple of 8.

But here’s the really amazing thing. In the Euclidean case there can be lots of different even unimodular lattices in a given dimension. In 8 dimensions there’s just one, up to isometry, called $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$. In 16 dimensions there are two. In 24 dimensions there are 24. In 32 dimensions there are at least 1,160,000,000, and the number continues to explode after that. On the other hand, in the Lorentzian case there’s just *one* even unimodular lattice in a given dimension, if there are any at all.

More precisely: given two even unimodular lattices in $<semantics>{\mathbb{R}}^{n+1,1}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^\{n+1,1\}</annotation></semantics>$, they are always isomorphic to each other via an **isometry**: a linear transformation that preserves the metric. We then call them **isometric**.

Let’s look at some examples. Up to isometry, $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$ is the only even unimodular lattice in 8-dimensional Euclidean space. We can identify it with the lattice of integral octonions, $<semantics>O\subseteq \mathbb{O}<annotation\; encoding="application/x-tex">\backslash mathbf\{O\}\; \backslash subseteq\; \backslash mathbb\{O\}</annotation></semantics>$, with the inner product

$$<semantics>g(X,X)=2X{X}^{*}<annotation\; encoding="application/x-tex">\; g(X,X)\; =\; 2\; X\; X^*</annotation></semantics>$$

$<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$ is usually called $<semantics>{E}_{10}<annotation\; encoding="application/x-tex">E\_\{10\}</annotation></semantics>$. Up to isometry, this is the unique even unimodular lattice in 10 dimensions. There are lots of ways to describe it, but last time we saw that it’s the lattice of $<semantics>2\times 2<annotation\; encoding="application/x-tex">2\; \backslash times\; 2</annotation></semantics>$ self-adjoint matrices with integral octonions as entries:

$$<semantics>{\U0001d525}_{2}(O)=\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{cc}a& X\\ {X}^{*}& b\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b\in \mathbb{Z},\phantom{\rule{thickmathspace}{0ex}}X\in O\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash mathfrak\{h\}\_2(\backslash mathbf\{O\})\; =\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{cc\}\; a\; \&\; X\; \backslash \backslash \; X^*\; \&\; b\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ;\; X\; \backslash in\; \backslash mathbf\{O\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

where the metric comes from $<semantics>-2<annotation\; encoding="application/x-tex">-2</annotation></semantics>$ times the determinant:

$$<semantics>x=\left(\begin{array}{cc}a& X\\ {X}^{*}& b\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\Rightarrow \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}g(x,x)=-\mathrm{det}(x)=2X{X}^{*}-2ab<annotation\; encoding="application/x-tex">\; x\; =\; \backslash left(\; \backslash begin\{array\}\{cc\}\; a\; \&\; X\; \backslash \backslash \; X^*\; \&\; b\; \backslash end\{array\}\; \backslash right)\; \backslash ;\backslash ;\; \backslash implies\; \backslash ;\backslash ;\; g(x,x)\; =\; -\; \backslash det(x)\; =\; 2\; X\; X^*\; -\; 2\; a\; b\; </annotation></semantics>$$

We’ll see a fancier formula like this later on.

There are 24 even unimodular lattices in 24-dimensional Euclidean space. One of them is

$$<semantics>{\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; </annotation></semantics>$$

Another is $<semantics>{\mathrm{D}}_{24}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_24</annotation></semantics>$. This is the lattice of vectors in $<semantics>{\mathbb{R}}^{24}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}^\{24\}</annotation></semantics>$ where the components are integers and their sum is even. It’s also the root lattice of the Lie group $<semantics>\mathrm{Spin}(48)<annotation\; encoding="application/x-tex">\backslash mathrm\{Spin\}(48)</annotation></semantics>$.

If we take the over-extension of any of these lattices, we get an even unimodular lattice in 26-dimensional Minkowski spacetime… and all these are isometric! The over-extension process ‘washes out the difference’ between them. In particular,

$$<semantics>{\mathrm{D}}_{24}^{++}\cong ({\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}{)}^{++}<annotation\; encoding="application/x-tex">\; \backslash mathrm\{D\}\_\{24\}^\{++\}\; \backslash cong\; (\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8)^\{++\}\; </annotation></semantics>$$

This is nice because up to a scale factor, $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$ is the lattice of integral octonions. So, there’s a description of $<semantics>{\mathrm{D}}_{24}^{++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{++\}</annotation></semantics>$ using three integral octonions! But the story is prettier if we go up an extra dimension.

### Very extended lattices

After the over-extended version $<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$of a lattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ in Euclidean space comes the ‘very extended’ version, called $<semantics>{L}^{+++}<annotation\; encoding="application/x-tex">L^\{+++\}</annotation></semantics>$. If you ponder the paper by Gaberdiel *et al*, you can see this is the direct sum of the over-extension $<semantics>{L}^{++}<annotation\; encoding="application/x-tex">L^\{++\}</annotation></semantics>$ and a 1-dimensional lattice called $<semantics>{\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{A\}\_1</annotation></semantics>$. $<semantics>{\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{A\}\_1</annotation></semantics>$ is just $<semantics>\mathbb{Z}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}</annotation></semantics>$ with the metric

$$<semantics>g(x,x)=2{x}^{2}<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; 2\; x^2\; </annotation></semantics>$$

It’s even but not unimodular.

In short, the **very extended** version of $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is

$$<semantics>{L}^{+++}={L}^{++}\oplus {\mathrm{A}}_{1}=L\oplus \mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">L^\{+++\}\; =\; L^\{++\}\; \backslash oplus\; \backslash mathrm\{A\}\_1\; =\; L\; \backslash oplus\; \backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1\; </annotation></semantics>$$

If $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is even, so is $<semantics>{L}^{+++}<annotation\; encoding="application/x-tex">L^\{+++\}</annotation></semantics>$. But if $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is unimodular, this will not be true of $<semantics>{L}^{+++}<annotation\; encoding="application/x-tex">L^\{+++\}</annotation></semantics>$.

The very extended version of $<semantics>{\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8</annotation></semantics>$ is called $<semantics>{\mathrm{E}}_{11}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{11\}</annotation></semantics>$. This a fascinating thing, but I want to talk about the very extended version of $<semantics>{\mathrm{D}}_{24}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}</annotation></semantics>$, and how to describe it using octonions.

Let $<semantics>{\U0001d525}_{3}(\mathbb{O})<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbb\{O\})</annotation></semantics>$ be the space of $<semantics>3\times 3<annotation\; encoding="application/x-tex">3\; \backslash times\; 3</annotation></semantics>$ self-adjoint octonionic matrices. It’s 27-dimensional since a typical element looks like

$$<semantics>x=\left(\begin{array}{ccc}a& X& Y\\ {X}^{*}& b& Z\\ {Y}^{*}& {Z}^{*}& c\end{array}\right)<annotation\; encoding="application/x-tex">\; x\; =\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; Y\; \backslash \backslash \; X^*\; \&\; b\; \&\; Z\; \backslash \backslash \; Y^*\; \&\; Z^*\; \&\; c\; \backslash end\{array\}\; \backslash right)\; </annotation></semantics>$$

where $<semantics>a,b,c\in \mathbb{R},X,Y,Z\in \mathbb{O}<annotation\; encoding="application/x-tex">a,b,c\; \backslash in\; \backslash mathbb\{R\},\; X,Y,Z\; \backslash in\; \backslash mathbb\{O\}</annotation></semantics>$. It’s called the **exceptional Jordan algebra**. We don’t need to know about Jordan algebras now, but this concept encapsulates the fact that if $<semantics>x\in {\U0001d525}_{3}(\mathbb{O})<annotation\; encoding="application/x-tex">x\; \backslash in\; \backslash mathfrak\{h\}\_3(\backslash mathbb\{O\})</annotation></semantics>$, so is $<semantics>{x}^{2}<annotation\; encoding="application/x-tex">x^2</annotation></semantics>$.

There’s a 2-parameter family of metrics on the exceptional Jordan algebra that are invariant under all Jordan algebra automorphisms. They have

$$<semantics>g(x,x)=\alpha tr({x}^{2})+\beta tr(x{)}^{2}<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; \backslash alpha\; \backslash tr(x^2)\; +\; \backslash beta\; \backslash tr(x)^2\; </annotation></semantics>$$

for $<semantics>\alpha ,\beta \in \mathbb{R}<annotation\; encoding="application/x-tex">\backslash alpha,\; \backslash beta\; \backslash in\; \backslash mathbb\{R\}</annotation></semantics>$ with $<semantics>\alpha \ne 0<annotation\; encoding="application/x-tex">\backslash alpha\; \backslash ne\; 0</annotation></semantics>$. Some are Euclidean and some are Lorentzian.

Sitting inside the exceptional Jordan algebra is the lattice of $<semantics>3\times 3<annotation\; encoding="application/x-tex">3\; \backslash times\; 3</annotation></semantics>$ self-adjoint matrices with integral octonions as entries:

$$<semantics>{\U0001d525}_{3}(O)=\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& X& Y\\ {X}^{*}& b& Z\\ {Y}^{*}& {Z}^{*}& c\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b,c\in \mathbb{Z},\phantom{\rule{thickmathspace}{0ex}}X,Y,Z\in O\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})\; =\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; Y\; \backslash \backslash \; X^*\; \&\; b\; \&\; Z\; \backslash \backslash \; Y^*\; \&\; Z^*\; \&\; c\; \backslash end\{array\}\; \backslash right)\; :\backslash ;\; a,b,c\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ;\; X,Y,Z\; \backslash in\; \backslash mathbf\{O\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

And here’s the cool part:

**Theorem.** There is a Lorentzian inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ on the exceptional Jordan algebra that is invariant under all automorphisms and makes the lattice $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ isometric to $<semantics>{\mathrm{K}}_{27}\cong {\mathrm{D}}_{24}^{+++}<annotation\; encoding="application/x-tex">\backslash mathrm\{K\}\_\{27\}\; \backslash cong\; \backslash mathrm\{D\}\_\{24\}^\{+++\}</annotation></semantics>$.

**Proof.** We will prove that the metric

$$<semantics>g(x,x)=tr({x}^{2})-tr(x{)}^{2}<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; \backslash tr(x^2)\; -\; \backslash tr(x)^2\; </annotation></semantics>$$

obeys all the conditions of this theorem. From what I’ve already said, it is invariant under all Jordan algebra automorphisms. The challenge is to show that it makes $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ isometric to $<semantics>{\mathrm{D}}_{24}^{+++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{+++\}</annotation></semantics>$. But instead of $<semantics>{\mathrm{D}}_{24}^{+++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{+++\}</annotation></semantics>$, we can work with $<semantics>({\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}{)}^{+++}<annotation\; encoding="application/x-tex">(\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8)^\{+++\}</annotation></semantics>$, since we have seen that

$$<semantics>{\mathrm{D}}_{24}^{+++}\cong ({\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}{)}^{+++}<annotation\; encoding="application/x-tex">\; \backslash mathrm\{D\}\_\{24\}^\{+++\}\; \backslash cong\; (\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8)^\{+++\}\; </annotation></semantics>$$

Let us examine the metric $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ in more detail. Take any element $<semantics>x\in {\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">x\; \backslash in\; \backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$:

$$<semantics>x=\left(\begin{array}{ccc}a& X& Y\\ {X}^{*}& b& Z\\ {Y}^{*}& {Z}^{*}& c\end{array}\right)<annotation\; encoding="application/x-tex">\; x\; =\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; Y\; \backslash \backslash \; X^*\; \&\; b\; \&\; Z\; \backslash \backslash \; Y^*\; \&\; Z^*\; \&\; c\; \backslash end\{array\}\; \backslash right)\; </annotation></semantics>$$

where $<semantics>a,b,c\in \mathbb{R},X,Y,Z\in \mathbb{O}<annotation\; encoding="application/x-tex">a,b,c\; \backslash in\; \backslash mathbb\{R\},\; X,Y,Z\; \backslash in\; \backslash mathbb\{O\}</annotation></semantics>$. Then

$$<semantics>\mathrm{tr}({x}^{2})={a}^{2}+{b}^{2}+{c}^{2}+2(X{X}^{*}+Y{Y}^{*}+Z{Z}^{*})<annotation\; encoding="application/x-tex">\; tr(x^2)\; =\; a^2\; +\; b^2\; +\; c^2\; +\; 2(X\; X^*\; +\; Y\; Y^*\; +\; Z\; Z^*)</annotation></semantics>$$

while

$$<semantics>\mathrm{tr}(x{)}^{2}=(a+b+c{)}^{2}<annotation\; encoding="application/x-tex">\; tr(x)^2\; =\; (a\; +\; b\; +\; c)^2\; </annotation></semantics>$$

Thus

$$<semantics>\begin{array}{ccl}g(x,x)& =& \mathrm{tr}({x}^{2})-\mathrm{tr}(x{)}^{2}\\ & =& 2(X{X}^{*}+Y{Y}^{*}+Z{Z}^{*})-2(ab+bc+ca)\end{array}<annotation\; encoding="application/x-tex">\; \backslash begin\{array\}\{ccl\}\; g(x,x)\; \&=\&\; tr(x^2)\; -\; tr(x)^2\; \backslash \backslash \; \&=\&\; 2(X\; X^*\; +\; Y\; Y^*\; +\; Z\; Z^*)\; -\; 2(a\; b\; +\; b\; c\; +\; c\; a)\; \backslash end\{array\}\; </annotation></semantics>$$

It follows that with this metric, the diagonal matrices are orthogonal to the off-diagonal matrices. An off-diagonal matrix $<semantics>x\in {\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">x\; \backslash in\; \backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ is a triple $<semantics>(X,Y,Z)\in {O}^{3}<annotation\; encoding="application/x-tex">(X,Y,Z)\; \backslash in\; \backslash mathbf\{O\}^3</annotation></semantics>$, and has

$$<semantics>g(x,x)=2(X{X}^{*}+Y{Y}^{*}+Z{Z}^{*})<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; 2(X\; X^*\; +\; Y\; Y^*\; +\; Z\; Z^*)\; </annotation></semantics>$$

Thanks to the factor of 2, this metric makes the lattice of these off-diagonal matrices isometric to $<semantics>{\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8</annotation></semantics>$. Since

$$<semantics>({\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}{)}^{+++}={\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus \mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\; (\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8)^\{+++\}\; =\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1\; </annotation></semantics>$$

it thus suffices to show that the 3-dimensional Lorentzian lattice of *diagonal* matrices in $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ is isometric to

$$<semantics>\mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\; \backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1\; </annotation></semantics>$$

A diagonal matrix $<semantics>x\in {\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">x\; \backslash in\; \backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ is a triple $<semantics>(a,b,c)\in {\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">(a,b,c)\; \backslash in\; \backslash mathbb\{Z\}^3</annotation></semantics>$, and on these triples the inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ is given by

$$<semantics>g(x,x)=-2(ab+bc+ca)<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; -2(a\; b\; +\; b\; c\; +\; c\; a)\; </annotation></semantics>$$

If we restrict attention to triples of the form $<semantics>x=(a,b,0)<annotation\; encoding="application/x-tex">x\; =\; (a,b,0)</annotation></semantics>$, we get a 2-dimensional Lorentzian lattice: a copy of $<semantics>{\mathbb{Z}}^{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}^2</annotation></semantics>$ with inner product

$$<semantics>g(x,x)=-2ab<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; -2a\; b</annotation></semantics>$$

This is just $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$.

We can use this to show that the lattice of *all* triples $<semantics>(a,b,c)\in {\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">(a,b,c)\; \backslash in\; \backslash mathbb\{Z\}^3</annotation></semantics>$, with the inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$, is isometric to $<semantics>\mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1</annotation></semantics>$.

Remember, $<semantics>{\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{A\}\_1</annotation></semantics>$ is a 1-dimensional lattice generated by a spacelike vector whose norm squared is 2. So, it suffices to show that the lattice $<semantics>{\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}^3</annotation></semantics>$ is generated by vectors of the form $<semantics>(a,b,0)<annotation\; encoding="application/x-tex">(a,b,0)</annotation></semantics>$ together with a spacelike vector of norm squared 2 that is orthogonal to all those of the form $<semantics>(a,b,0)<annotation\; encoding="application/x-tex">(a,b,0)</annotation></semantics>$.

To do this, we need to describe the inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ on $<semantics>{\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}^3</annotation></semantics>$ more explicitly. For this, we can use polarization identity

$$<semantics>g(x,x\prime )=\frac{1}{2}(g(x+x\prime ,x+x\prime )-g(x,x)-g(x\prime ,x\prime ))<annotation\; encoding="application/x-tex">\; g(x,x\text{\'})\; =\; \backslash frac\{1\}\{2\}(\; g(x+x\text{\'},x+x\text{\'})\; -\; g(x,x)\; -\; g(x\text{\'},x\text{\'}))\; </annotation></semantics>$$

Remember, if $<semantics>x=(a,b,c)<annotation\; encoding="application/x-tex">x\; =\; (a,b,c)</annotation></semantics>$ we have

$$<semantics>g(x,x)=-2(ab+bc+ca)<annotation\; encoding="application/x-tex">\; g(x,x)\; =\; -2(a\; b\; +\; b\; c\; +\; c\; a)\; </annotation></semantics>$$

So, if we also have $<semantics>x\prime =(a\prime ,b\prime ,c\prime )<annotation\; encoding="application/x-tex">x\text{\'}\; =\; (a\text{\'},b\text{\'},c\text{\'})</annotation></semantics>$, the polarization identity gives

$$<semantics>g(x,x\prime )=-(ab\prime +a\prime b)-(bc\prime +bc\prime )-(ca\prime +c\prime a)<annotation\; encoding="application/x-tex">\; g(x,x\text{\'})\; =\; -(a\; b\text{\'}+a\text{\'}\; b)\; -\; (b\; c\text{\'}+\; b\; c\text{\'})\; -\; (c\; a\text{\'}\; +\; c\text{\'}a)</annotation></semantics>$$

We are looking for a spacelike vector $<semantics>x\prime =(a\prime ,b\prime ,c\prime )<annotation\; encoding="application/x-tex">x\text{\'}\; =\; (a\text{\'},b\text{\'},c\text{\'})</annotation></semantics>$ that is orthogonal to all those of the form $<semantics>x=(a,b,0)<annotation\; encoding="application/x-tex">x\; =\; (a,b,0)</annotation></semantics>$. For this, it is necessary and sufficient to have

$$<semantics>0=g((1,0,0),(a\prime ,b\prime ,c\prime ))=-b\prime -c\prime <annotation\; encoding="application/x-tex">\; 0\; =\; g((1,0,0),(a\text{\'},b\text{\'},c\text{\'}))\; =\; -\; b\text{\'}\; -\; c\text{\'}\; </annotation></semantics>$$

and

$$<semantics>0=g((0,1,0),(a\prime ,b\prime ,c\prime ))=-a\prime -c\prime <annotation\; encoding="application/x-tex">\; 0\; =\; g((0,1,0),\; (a\text{\'},b\text{\'},c\text{\'}))\; =\; -\; a\text{\'}\; -\; c\text{\'}\; </annotation></semantics>$$

An example is $<semantics>x\prime =(1,1,-1)<annotation\; encoding="application/x-tex">x\text{\'}\; =\; (1,1,-1)</annotation></semantics>$. This has

$$<semantics>g(x\prime ,x\prime )=-2(1-1-1)=2<annotation\; encoding="application/x-tex">\; g(x\text{\'},x\text{\'})\; =\; -2(1\; -\; 1\; -\; 1)\; =\; 2\; </annotation></semantics>$$

so it is spacelike, as desired. Even better, it has norm squared two. And even better, this vector $<semantics>x\prime <annotation\; encoding="application/x-tex">x\text{\'}</annotation></semantics>$, along with those of the form $<semantics>(a,b,0)<annotation\; encoding="application/x-tex">(a,b,0)</annotation></semantics>$, generates the lattice $<semantics>{\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}^3</annotation></semantics>$.

So we have shown what we needed: the lattice of all triples $<semantics>(a,b,c)\in {\mathbb{Z}}^{3}<annotation\; encoding="application/x-tex">(a,b,c)\; \backslash in\; \backslash mathbb\{Z\}^3</annotation></semantics>$ is generated by those of the form $<semantics>(a,b,0)<annotation\; encoding="application/x-tex">(a,b,0)</annotation></semantics>$ together with a spacelike vector with norm squared 2 that is orthogonal to all those of the form $<semantics>(a,b,0)<annotation\; encoding="application/x-tex">(a,b,0)</annotation></semantics>$. $<semantics>\blacksquare <annotation\; encoding="application/x-tex">\backslash blacksquare</annotation></semantics>$

This theorem has three nice spinoffs:

**Corollary.** With the same Lorentzian inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ on the exceptional Jordan algebra, the lattice $<semantics>{\mathrm{D}}_{24}^{++}<annotation\; encoding="application/x-tex">\backslash mathrm\{D\}\_\{24\}^\{++\}</annotation></semantics>$ is isometric to the sublattice of $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ where a fixed diagonal entry is set equal to zero, e.g.:

$$<semantics>\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& X& Y\\ {X}^{*}& b& Z\\ {Y}^{*}& {Z}^{*}& 0\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b\in \mathbb{Z},\phantom{\rule{thickmathspace}{0ex}}X,Y,Z\in O\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; Y\; \backslash \backslash \; X^*\; \&\; b\; \&\; Z\; \backslash \backslash \; Y^*\; \&\; Z^*\; \&\; 0\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ;\; X,Y,Z\; \backslash in\; \backslash mathbf\{O\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

**Proof.** Use the fact that with the metric $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$, the diagonal matrices

$$<semantics>\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& 0\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b\in \mathbb{Z}\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; b\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b\; \backslash in\; \backslash mathbb\{Z\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

form a copy of $<semantics>\mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}</annotation></semantics>$, so the matrices above form a copy of

$$<semantics>{\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus \mathrm{H}\cong ({\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}\oplus {\mathrm{E}}_{8}{)}^{++}\cong {\mathrm{D}}_{24}^{++}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\blacksquare <annotation\; encoding="application/x-tex">\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{H\}\; \backslash cong\; (\backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{E\}\_8)^\{++\}\; \backslash cong\; \backslash mathrm\{D\}\_\{24\}^\{++\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash blacksquare\; </annotation></semantics>$$

**Corollary.** With the same Lorentzian inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ on the exceptional Jordan algebra, the lattice $<semantics>{\mathrm{E}}_{11}={E}_{8}^{+++}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{11\}\; =\; E\_8^\{+++\}</annotation></semantics>$ is isometric to the sublattice of $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ where two fixed off-diagonal entries are set equal to zero, e.g.:

$$<semantics>\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& X& 0\\ {X}^{*}& b& 0\\ 0& 0& c\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b,c\in \mathbb{Z},\phantom{\rule{thickmathspace}{0ex}}X\in O\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; 0\; \backslash \backslash \; X^*\; \&\; b\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; c\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b,c\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ;\; X\backslash in\; \backslash mathbf\{O\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

**Proof.** Use the fact that with the metric $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$, the diagonal matrices

$$<semantics>\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b\in \mathbb{Z}\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; b\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; c\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b\; \backslash in\; \backslash mathbb\{Z\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

form a copy of $<semantics>\mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1</annotation></semantics>$, so the matrices above form a copy of

$$<semantics>{\mathrm{E}}_{8}\oplus \mathrm{H}\oplus {\mathrm{A}}_{1}\cong {\mathrm{E}}_{8}^{+++}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\blacksquare <annotation\; encoding="application/x-tex">\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1\; \backslash cong\; \backslash mathrm\{E\}\_8^\{+++\}\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash blacksquare\; </annotation></semantics>$$

**Corollary.** With the same Lorentzian inner product $<semantics>g<annotation\; encoding="application/x-tex">g</annotation></semantics>$ on the exceptional Jordan algebra, the lattice $<semantics>{\mathrm{E}}_{10}={\mathrm{E}}_{8}^{++}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_\{10\}\; =\; \backslash mathrm\{E\}\_8^\{++\}</annotation></semantics>$ is isometric to the sublattice of $<semantics>{\U0001d525}_{3}(O)<annotation\; encoding="application/x-tex">\backslash mathfrak\{h\}\_3(\backslash mathbf\{O\})</annotation></semantics>$ where two fixed off-diagonal entries and one diagonal entry are set equal to zero, e.g.:

$$<semantics>\{\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}a& X& 0\\ {X}^{*}& b& 0\\ 0& 0& 0\end{array}\right):\phantom{\rule{thickmathspace}{0ex}}a,b,c\in \mathbb{Z},\phantom{\rule{thickmathspace}{0ex}}X\in O\phantom{\rule{thickmathspace}{0ex}}\}<annotation\; encoding="application/x-tex">\; \backslash left\backslash \{\; \backslash ;\; \backslash left(\; \backslash begin\{array\}\{ccc\}\; a\; \&\; X\; \&\; 0\; \backslash \backslash \; X^*\; \&\; b\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 0\; \backslash end\{array\}\; \backslash right)\; :\; \backslash ;\; a,b,c\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ;\; X\backslash in\; \backslash mathbf\{O\}\; \backslash ;\; \backslash right\backslash \}\; </annotation></semantics>$$

**Proof.** Use the previous corollary; this is the obvious copy of $<semantics>{\mathrm{E}}_{8}^{++}\cong {\mathrm{E}}_{8}\oplus \mathrm{H}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8^\{++\}\; \backslash cong\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{H\}</annotation></semantics>$ inside $<semantics>{\mathrm{E}}_{8}^{+++}\cong {\mathrm{E}}_{8}\oplus \mathrm{H}\oplus {\mathrm{A}}_{1}<annotation\; encoding="application/x-tex">\backslash mathrm\{E\}\_8^\{+++\}\; \backslash cong\; \backslash mathrm\{E\}\_8\; \backslash oplus\; \backslash mathrm\{H\}\; \backslash oplus\; \backslash mathrm\{A\}\_1</annotation></semantics>$. $<semantics>\blacksquare <annotation\; encoding="application/x-tex">\backslash blacksquare</annotation></semantics>$