  We give a classification of Gorenstein Fano bi-equivariant compactifications
of semisimple complex Lie groups with rank two, and determine which of them are
equivariant K-stable and admit (singular) K\"{a}hler-Einstein metrics. As a
consequence, we obtain several explicit examples of K-stable Fano varieties
admitting (singular) K\"{a}hler-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 K\"{a}hler-Ricci
flow is of type II.

Differential Geometry;Algebraic Geometry;Complex Variables