I am working with biquotients and positive curvatures and I was able to give a relatively simple proof for the following:
Theorem: Let $G$ be a compact connected Lie group with a bi-invariant metric $Q$. Let $H\times K\subset G\times G$ where $H,K$ are closed subgroups of $G$. Consider the action $$(h,k)\cdot g := hgk^{-1}$$ and assume it is free. Then, if $G$ is semi-simple the quotient $H\setminus G/K$ has positive Ricci curvature on the submersion metric.
The proof uses some properties of semi-simple Lie groups such as the its decomposition as direct sums of ideals and the fact it is centerless.
My questions are: is this result stated in this way well known in the literature? Furthermore, how is the hypothesis of $G$ being semi-simple related to Lorenz Schwachhoefer, Wilderich Tuschmann theorems on positive Ricci curvature of quotients where positive Ricci curvature is ensured (if only if condition) with $H\setminus G/K$ having finite fundamental group?
