# $\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 $$M = S^3 \times S^3$$. Then the cone metric can be written as $$g = dt^2 + g_t$$, where $$g_t = \sum_{i=1}^3(2A_i)^2\eta_i^+ \wedge \eta_i^+ + (2B_i)^2\eta_i^- \wedge \eta_i^-$$ is the compatible metric given by the $$\SU(2)\times\SU(2)$$ invariant $$\SU(3)$$ structure on $$\{t\}\times M$$.

Here $$\eta_i^\pm$$ are the standard coframe of 1-forms, while the functions $$A_i(t)$$, $$B_i(t)$$ specify the deformation of the cone singularity. I have the following questions:

1. How can we see that the metric $$g_t$$ (and in general the $$\SU(3)$$ structure as given) is $$\SU(2)\times\SU(2)$$ invariant?

2. Why does an extra $$\operatorname U(1)$$ symmetry force $$A_2=A_3$$, $$B_2=B_3$$?

3. Why does an extra $$\SU(2)$$ symmetry force $$A_1=A_2=A_3$$, $$B_1=B_2=B_3$$?

$$\DeclareMathOperator\SU{SU}$$For the first one, you should note that the six $$1$$-forms $$\eta^{\pm}_i$$ are all defined using the left-invariant $$1$$-forms on $$S^3 \times S^3 \cong \SU(2) \times \SU(2)$$, hence they are $$\SU(2)^2$$-invariant. Since $$g_t$$, $$\omega$$ and $$\Omega_j$$ are all defined using $$\eta^{\pm}_i$$, it follows that the $$\SU(3)$$-structure is invariant by $$\SU(2)^2$$.
For the second one, the $$\operatorname U(1)$$ action is generated by the vector field $$T_1^+$$. If you impose that $$\mathcal{L}_{T_1^+}g_t=0$$ then you should find that $$A_2=A_3$$ and $$B_2=B_3$$. For the third one, just repeat the same argument with $$T_2^+$$ and $$T_3^+$$ as well.