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 $G=\mathrm{SU}(2)\times\mathrm{SU}(2)\cong S^3\times S^3$. 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 $σ$ with $\trσ=-2$ 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 $g_{\rm can}$ or the Jensen nearly Kähler metric $g_{\rm NK}$.