Second proof for SU(2) isomorphic to LB1 X LB1 X LB1 theorem

TL;DR

This paper provides a second independent proof of the isomorphism between SU(2) and LB1 X LB1 X LB1, using a novel approach involving fixed electromagnetic tetrads and a single SU(2) tetrad.

Contribution

It introduces an alternative proof method for the SU(2) isomorphism, reversing the choice of tetrads and fixing a single SU(2) tetrad with different electromagnetic tetrads.

Findings

Confirmed the isomorphism using a new tetrad approach

Established equivalence via alternative proof method

Reinforced the relationship between SU(2) and LB groups

Abstract

In this note we present a second independent proof for the theorem introduced previously that establishes an isomorphism between SU(2) and LB1 X LB1 X LB1. Since the local groups LB1 and LB2 are isomorphic, it was also previously proved a similar result for LB2 X LB2 X LB2. We are going to reverse the three sets of tetrads that are going to be used in order to prove this new version. Instead of choosing three SU(2) different tetrads keeping the electromagnetic tetrad the same for all three sets of SU(2) tetrads, we are going to keep fixed the SU(2) tetrad, that is, we are going to pick just one SU(2) tetrad but choose three different arbitrary but fixed electromagnetic tetrads in order to gauge locally the only local SU(2) tetrad involved in our new version of this theorem. The same result will be obtained via an alternative but equivalent way.