Rotations in the Space of Split Octonions

TL;DR

This paper explores the geometry of split octonions, illustrating their transformations and symmetries, including the invariant group SO(4,4) and the relation to Lorentz transformations in classical physics.

Contribution

It introduces a modified Fano graphic for split octonions and clarifies the distinction between active and passive transformations in octonionic 8-space.

Findings

Passive transformations form the SO(4,4) group.

Active rotations involve O(3,4)-boosts and G2.

Transformations reduce to Lorentz group in classical limit.

Abstract

The geometrical application of split octonions is considered. The modified Fano graphic, which represents products of the basis units of split octonionic, having David's Star shape, is presented. It is shown that active and passive transformations of coordinates in octonionic '8-space' are not equivalent. The group of passive transformations that leave invariant the norm of split octonions is SO(4,4), while active rotations is done by the direct product of O(3,4)-boosts and real non-compact form of the exceptional group $G_2$. In classical limit these transformations reduce to the standard Lorentz group.