Euler angles for G2

TL;DR

This paper introduces a new Euler-like parametrization for the exceptional Lie group G2, highlighting its fibration structure and providing explicit formulas for the Haar measure and related geometric structures.

Contribution

It presents a simple, Euler-angle inspired parametrization of G2, elucidating its fibration over quaternionic subalgebras and deriving explicit measures and metrics.

Findings

Explicit Euler parametrization for G2

Derived Haar measure expression for G2

Constructed Einstein metric for the base space H

Abstract

We provide a simple parametrization for the group G2, which is analogous to the Euler parametrization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.