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$\DeclareMathOperator\SU{SU}$$\SU(2)\times \SU(2)$ invariant $\SU(3)$-structure on $\{t\} \times M^6$

$\DeclareMathOperator\SU{SU}$I am reading Jason Lotay and Goncalo Oliveira's paper $\SU(2)^2$ invariant $G_2$-instantons, and have a few questions from the same. If we consider the space $M = S^3 \...