$\mathcal{N}=3$ supersymmetric AdS${}_4$ solutions in $d=11$ supergravity

We derive the necessary and sufficient conditions for AdS${}_4$ solutions of $d=11$ supergravity to preserve $\mathcal{N}\le 3$ supersymmetry. We use our setup to first impose $\mathcal{N}=2$ supersymmetry, recovering a previously known class of solutions, providing a non-trivial consistency of our equations. We then focus on the $\mathcal{N}=3$ case, first deriving the Tri-Sasaki class and then proving its uniqueness as the only $\mathcal{N}=3$ AdS${}_4$ class in eleven dimensional supergravity. Our work proves that all supersymmetric AdS${}_4$ solutions preserving $\mathcal{N}>2$ supersymmetry are necessarily of Tri-Sasaki type.