The classification of $SU(2)^2$ biquotients of rank $3$ Lie groups

TL;DR

This paper classifies all compact simply connected biquotients formed by specific rank 3 Lie groups, identifying the exact number of inhomogeneous biquotients in each case.

Contribution

It provides a complete classification of biquotients of the form G//SU(2)^2 for certain rank 3 Lie groups, including counts of inhomogeneous examples.

Findings

2 inhomogeneous biquotients for G=SU(4) and G=G2×SU(2)

10 inhomogeneous biquotients for G=SO(7) and Spin(7)

Complete classification of these biquotients

Abstract

We classify all compact simply connected biquotients of the form $G/\!\!/ SU(2)^2$ for $G =SU(4), SO(7), Spin(7)$, or $G = \mathbf{G}_2\times SU(2)$. In particular, we show there are precisely $2$ inhomogeneous reduced biquotients in the first and last case, and $10$ in the middle cases.