# Geometry of the Non-Compact G(2)

## TL;DR

This paper explores the geometry of the non-compact form of G(2), revealing its role in generating rotations in a 7D Minkowski space with unique translation properties and relating its Casimir element to Lorentz and Poincare groups.

## Contribution

It introduces the geometrical structure of non-compact G(2), including its Casimir operator, and connects it to known spacetime symmetries in a novel way.

## Key findings

- Non-compact G(2) generates specific rotations in 7D Minkowski space.
- Space-time translations are non-commutative and linked to extra time-like coordinates.
- Casimir element of non-compact G(2) expressed via Lorentz and Poincare Casimirs.

## Abstract

Geometrical applications of the non-compact form of Cartan's exceptional Lie group G(2) is considered. This group generates specific rotations of 7-dimensional Minkowski-like space with three extra time-like coordinates and in some limiting cases imitates standard Poincare transformations. In this model space-time translations are non-commutative and are represented by the rotations towards the extra time-like coordinates. The second order Casimir element of non-compact G(2) and its expression by the Casimir operators of the Lorentz and Poincare groups are found.