Integral Octonions, Octonion XY-Product, and the Leech Lattice

TL;DR

This paper explores the structure of integral octonions derived from the XY-product, draws parallels with quaternion Z-product, and links these algebraic constructs to important laminated lattices including the Leech lattice.

Contribution

It introduces a novel perspective connecting octonion XY-product and quaternion Z-product to the structure of laminated lattices, including the Leech lattice.

Findings

Integral octonions relate to XY-product structure.

Connections established between octonion products and laminated lattices.

Insights into the algebraic underpinnings of the Leech lattice.

Abstract

The integral octonions arise from the octonion XY-product. A parallel is shown to exist with the quaternion Z-product. Connections to the laminated lattices, $\Lambda_{4}$, $\Lambda_{8}$, $\Lambda_{16}$ and $\Lambda_{24}$ (Leech), are developed.