  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 $SO_4(\mathbb C)$ which admit a
K\"ahler-Einstein metric with the same volume as that of a smooth Fano
compactification of $SO_4(\mathbb C)$.
