A complete classification of left-invariant Einstein metrics on $S^3\times S^3$

We complete the classification of compact simply connected homogeneous Einstein manifolds in dimension six by resolving the remaining cases for left-invariant Einstein metrics on $G=\mathrm{S}\mathrm{U}(2)\times \mathrm{S}\mathrm{U}(2)\cong S^3\times S^3$. Previous work leaves two cases for the isotropy group $K$, namely $K=\{e\}$ and $K\cong {\mathbb{Z}}_2$. We first rule out $K=\{e\}$. Then we show that any ${\mathbb{Z}}_2$ symmetry generated by an inner involution $\sigma$ with $\text{tr}\sigma =-2$ necessarily extends to a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry. Together with the previously known classification results, these theorems imply that every left-invariant Einstein metric on $G$ is, up to homothety and isometry, either the standard product metric $g_{\mathrm{c}\mathrm{a}\mathrm{n}}$ or the Jensen nearly Kähler metric $g_{\mathrm{N}\mathrm{K}}$.