K-stability of Gorenstein Fano group compactifications with rank two

We give a classification of Gorenstein Fano bi-equivariant compactifications of semi-simple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler-Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler-Ricci flow is of type II.