Let's say we have a homogeneous space $H\backslash G$.


Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure? 


I am particularly interested in a situation when $H\backslash G$ is maximally-noncompact, i.e. $H$ is a maximally compact subgroup of $G$.


I hope, my question does not sound too broad. Maybe this question has a trivial answer, but from a background of a theoretical physicist, it is not obvious.



Conformally flat homogeneous spaces
A complete classification of homogeneous conformally flat Riemannian manifolds is given  [here](https://link.springer.com/article/10.1007/BF01812986), namely, [Alekseevskiĭ, D. V.; Kimelʹfelʹd, B. N. Classification of homogeneous conformally flat Riemannian manifolds. Mat. Zametki 24 (1978), no. 1, 103–110, 143]. 


In the simply-connected case the list consists of the Euclidean space $\mathbb R^n$, the hyperbolic space $H^n$, the round sphere $S^n$, and the products $H^{n-1}\times \mathbb R$ and $H^{n-k}\times S^k$. They also give a longer list in the non-simply-connected case, which includes e.g. any flat torus.


