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Integral elements of Okubo algebra and the E8-lattice

This paper investigates the arithmetic properties of the Coxeter-Dickson $E_8$ -order under para-octonionic and Okubo products, demonstrating that while the former preserves the $E_8$ lattice over $\mathbb{Z}$ , the latter necessitates a $\mathbb{Z}[\sqrt{3}]$ -order and requires 2-adic saturation to recover the $E_8$ lattice.

Original authors: Daniele Corradetti

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniele Corradetti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a perfect, symmetrical city called E8. In the world of mathematics, this city is a famous "lattice"—a grid of points in eight-dimensional space that is incredibly efficient and beautiful. For a long time, mathematicians knew how to build this city using a specific set of rules based on a number system called Octonions. This system is like a "perfect blueprint" where the numbers (called integers) fit together perfectly to create the E8 city.

This paper explores what happens when we try to build the same city using two different, slightly "twisted" versions of those rules. The author, Daniele Corradetti, asks: If we change the rules of multiplication, does the E8 city still stand?

Here is the story of the findings, broken down into simple concepts:

1. The Three Types of "Number Cities"

To understand the experiment, you need to know there are three different ways to do math in this 8-dimensional space:

The Octonions (The Original): The classic, standard way. It has a "1" (a unit) that works like normal multiplication. This builds the perfect E8 city.
The Para-Octonions (The Mirror): This is like looking at the original city in a mirror. The rules are slightly flipped, but the "1" is still there (though it acts a bit differently).
The Okubo Algebra (The Puzzle): This is the tricky one. It doesn't have a "1" at all. Instead, it has "idempotents" (numbers that stay the same when multiplied by themselves). It's like a city built without a central square.

2. The First Test: The Mirror City (Para-Octonions)

The author first checked the Para-Octonions.

The Result: It worked perfectly!
The Analogy: Imagine you have a set of Lego bricks that build the E8 city. You decide to flip the bricks over (the "mirror" rule). Surprisingly, the bricks still snap together perfectly. The same E8 city stands tall. The "integers" (the bricks) still fit the grid exactly as they did before.
Conclusion: The E8 lattice is robust enough to handle this mirror version.

3. The Second Test: The Puzzle City (Okubo Algebra)

Next, the author tried to build the city using the Okubo Algebra. This is where things get messy.

The Problem: When you try to multiply the standard "bricks" (integers) using the Okubo rules, the pieces start to break. The math requires numbers like $\sqrt{3}$ (the square root of 3) to work.
The Analogy: Imagine trying to build your E8 city with standard square bricks, but the new rules demand that you cut the bricks in half or use triangular pieces. If you try to force the standard square bricks together, they don't fit; they leave gaps or stick out. The original E8 city collapses under these new rules.
The "Obstruction": The paper calls this the " $\sqrt{3}$ -obstruction." The standard integer grid just isn't compatible with the Okubo multiplication.

4. The Fix: Rescaling the Bricks

The author didn't give up. He asked: Can we fix the bricks so they fit the Okubo rules?

The Solution: Yes, but we have to change the size of the bricks. The author found a specific "scaling" trick (multiplying some dimensions by 2 and others by 4).
The Result: After resizing the bricks, they do fit together. However, they don't build the original E8 city. They build a smaller, denser version of the city inside the E8 grid.
The "Shadow": The author calls this smaller version the "metric shadow." It is a sub-city (a sublattice) that sits inside the big E8 city. It is a "conductor sublattice," which is a fancy way of saying it's a specific, smaller grid that guides the structure but isn't the whole thing.

5. Recovering the Original City

So, we have a smaller city (the shadow) built with the Okubo rules. Can we get the big E8 city back?

The Catch: You can get the big E8 city back, but not by multiplying numbers. You have to "glue" the pieces together or "fill in the gaps" (a process called saturation).
The Meaning: The Okubo rules create a structure that contains the E8 city, but only if you look at it through a mathematical microscope (2-adic saturation). If you try to multiply numbers within that structure, the E8 city breaks again.
The Takeaway: The connection between the Okubo algebra and the E8 lattice is arithmetic (about how the numbers fit together) rather than multiplicative (about how the numbers multiply).

Summary

Para-Octonions: The E8 lattice is happy. It stays exactly the same.
Okubo Algebra: The E8 lattice breaks. You have to shrink it and change the rules to make it work.
The Big Picture: The E8 lattice is the "gold standard" of these 8-dimensional structures. The paper proves that while the Okubo algebra is related to E8, it doesn't build E8 directly. Instead, it builds a smaller, hidden version of E8 that can only be fully revealed by "gluing" the pieces together, not by simple multiplication.

In short, the paper shows that the famous E8 lattice is very flexible with some rule changes, but the Okubo algebra requires a completely different kind of foundation—one that is mathematically related to E8, but not identical to it.

Technical Summary: Integral Elements of Okubo Algebra and the $E_8$ -Lattice

Problem Statement
The paper addresses the definition and structural properties of integral elements within eight-dimensional real division algebras, specifically extending the classical theory of integral octonions to non-unital algebras. While the integral structure of the unital octonions ( $O$ ) is well-established via the Coxeter-Dickson order ( $OE_8$ ), which forms the $E_8$ lattice, the status of integral elements in the para-octonions ($pO$) and the real Okubo algebra ( $O$ ) remains less systematic. Standard definitions of integral elements (e.g., by Dickson and refined by Johnson) rely on the existence of a unit element, a condition not met by $pO$ (which has a para-unit) or $O$ (which is non-unital and possesses only idempotents). The central problem is to determine whether the Coxeter-Dickson order $OE_8$ remains closed under the products of these alternative algebras and, if not, to characterize the resulting arithmetic structures.

Methodology
The author employs a comparative algebraic and arithmetic approach:

Algebraic Review: The paper reviews the three 8-dimensional composition algebras: the unital octonions ( $O$ ), the para-unital para-octonions ($pO$), and the non-unital Okubo algebra ( $O$ ). It establishes the relationships between their products, noting that $pO$ and $O$ can be derived from $O$ via specific product modifications (involving conjugation and an order-three automorphism $\tau$ , respectively).
Adaptation of Integral Definitions: The author adapts Johnson's definition of integral systems for non-unital algebras. This involves replacing the unit element with a chosen idempotent (the para-unit for $pO$ and a specific idempotent $e$ for $O$ ) and defining a relative trace functional.
Closure Testing: The Coxeter-Dickson order $OE_8$ is tested for closure under the para-octonionic product ( $x \bullet y = \bar{x} \cdot \bar{y}$ ) and the Okubo (Petersson) product ( $x * y = \tau(x) \cdot \tau^2(y)$ ).
Arithmetic Scaling: When closure fails for the Okubo product due to coefficients in $\mathbb{Q}(\sqrt{3})$ , the author constructs a scaled basis using a diagonal matrix $D = \text{diag}(2, 2, 2, 2, 4, 4, 4, 4)$ to define a new order over the ring $R = \mathbb{Z}[\sqrt{3}]$ .
Lattice Analysis: The paper analyzes the "metric shadow" (the $\mathbb{Z}$ -span of the scaled basis) to determine its relationship with the standard $E_8$ lattice, calculating indices, determinants, and minimal norms. It further investigates the recovery of $E_8$ via 2-adic saturation and gluing.

Key Contributions and Results

Para-Octonionic Closure: The paper proves that the Coxeter-Dickson order $OE_8$ is closed under the para-octonionic product. Consequently, $OE_8$ constitutes a genuine $\mathbb{Z}$ -integral system for $pO$, and its underlying lattice remains the exceptional $E_8$ . This result mirrors the classical octonionic case.
The $\sqrt{3}$ -Obstruction: For the Okubo algebra, the paper demonstrates that $OE_8$ is not closed under the Petersson product as a $\mathbb{Z}$ -module. The product of integral basis elements introduces coefficients involving $\sqrt{3}$ , specifically showing that the coefficient field is $K = \mathbb{Q}(\sqrt{3})$ .
Scaled $\mathbb{Z}[\sqrt{3}]$ -Order: By applying a specific diagonal 2-adic scaling, the author constructs a closed order $\mathcal{O}_0$ over the ring $R = \mathbb{Z}[\sqrt{3}]$ . This order is closed under the Okubo product, and its relative trace, norm, and bilinear form take values in $R$ .
The Conductor Sublattice: The direct $\mathbb{Z}$ $Z$ -span of the scaled basis (the "metric shadow"), denoted $L_{Ok}$ $L_{O k}$ , is identified as a sublattice of $E_8$ $E_{8}$ . Crucially, $L_{Ok}$ $L_{O k}$ is not isometric to $E_8$ $E_{8}$ . It is a conductor sublattice with:
- Index $[E_8 : L_{Ok}] = 2^{12}$ .
- Determinant $\det(L_{Ok}) = 2^{24}$ .
- Minimum norm $\min(L_{Ok}) = 8$ (meaning it contains no roots of norm 2, unlike $E_8$ ).
Recovery of $E_8$ : The paper establishes that the $E_8$ lattice can be recovered from $L_{Ok}$ only through 2-adic saturation or by gluing along a maximal isotropic subgroup. However, this recovery is purely metric-arithmetic; the recovered $E_8$ lattice is not closed under the Okubo product.

Significance and Claims
The paper claims to provide the first explicit characterization of the integral relation between the Coxeter-Dickson order and the Okubo algebra. Its primary significance lies in distinguishing three distinct layers that are often conflated:

The classical Euclidean lattice $E_8$ .
The Coxeter-Dickson octonionic order over $\mathbb{Z}$ .
The Okubo arithmetic order, which naturally forces an extension to $\mathbb{Z}[\sqrt{3}]$ .

The author emphasizes that the Okubo integral structure does not form the $E_8$ lattice over $\mathbb{Z}$ . Instead, it generates a 2-primary conductor sublattice. The recovery of $E_8$ from this sublattice is a metric operation (saturation/gluing) rather than a multiplicative one, as the saturated lattice loses closure under the Okubo product. This distinction highlights a fundamental arithmetic obstruction in the Okubo case that is absent in the para-octonionic case.

The paper concludes modestly, noting that while the arithmetic stabilizers and finite orbits of these structures are identified, the full arithmetic automorphism group remains to be fully determined, and the relationship between these conductor lattices and the geometry of the Gosset polytope warrants further investigation.