Left Invariance of the 4D Cross Product and the Decomposition of SO(4)

The 4D cross product A×B, defined by the fourth quadrant of the octonion multiplication table, is Im(ĀB) in quaternion form. It is invariant under left multiplication by unit quaternions and covariant, as a 3D rotation, under right multiplication. The second cross product (A×B)′ = Im(AB̄) is, conversely, right-invariant and left-covariant. From this left invariance the following results follow at once.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-26
DOI
https://doi.org/10.5281/zenodo.23657068
Primary Topic
Algebraic and Geometric Analysis
Type
article
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article

Left Invariance of the 4D Cross Product and the Decomposition of SO(4)

Sung-tae Lim
1 links
Zenodo (CERN European Organization for Nuclear Research)
Algebraic and Geometric Analysis
article

Left Invariance of the 4D Cross Product and the Decomposition of SO(4)

Sung-tae Lim
article en

Abstract

The 4D cross product A×B, defined by the fourth quadrant of the octonion multiplication table, is Im(ĀB) in quaternion form. It is invariant under left multiplication by unit quaternions and covariant, as a 3D rotation, under right multiplication. The second cross product (A×B)′ = Im(AB̄) is, conversely, right-invariant and left-covariant. From this left invariance the following results follow at once.

Zenodo (CERN European Organization for Nuclear Research)
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Algebraic and Geometric Analysis
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Left Invariance of the 4D Cross Product and the Decomposition of SO(4) — Sung-tae Lim · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS