Singular K\"ahler-Einstein metrics on $\mathbb{Q}$-Fano compactifications of Lie groups

In this paper, we prove an existence result for K"ahler-Einstein metrics on $\mathbb{Q}$-Fano compactifications of Lie groups. As an application, we classify $\mathbb{Q}$-Fano compactifications of $S{O}_4(\mathbb{C})$ which admit a K"ahler-Einstein metric with the same volume as that of a smooth Fano compactification of $S{O}_4(\mathbb{C})$.