I am trying to remember a result stating that under certain assumptions, 
given a transitive smooth action of a (compact?) Lie group on a smooth manifold, also the action of the semisimple factor is transitive. It could be due to Borel, but I cannot remember or find the reference. 


Does anybody know? 


Edit: the manifold must be compact and simply-connected. What else?



The role of semisimple factors in transitive group actions on manifolds
Yes, it's true: whenever $G$ is a compact Lie group and $X$ a simply connected (topological) manifold, and $G$ acts transitively continuously on $X$, then $G'=[G^0,G^0]$ acts transitively.


Indeed, one can write $X=G/H$. The $G^0$-orbits being open, $G^0$ acts transitively, so we can suppose that $G$ is connected. We have a fibre bundle $X=G/H\to G/HG'$, with connected fiber $HG'/H=G'/(H\cap G')$


Since fibers are connected, this induces an exact sequence $\pi\_1(X)\to \pi\_1(G/HG')\to 1$. Hence the torus $G/HG'$ is simply connected, which forces it to be a singleton. So $HG'=G$. This means that $G'$ acts transitively on $G/H=X$.


