TITLE: A bounded homogeneous space which fails to be symmetric?
QUESTION [6 upvotes]: Do we have examples of a contractible bounded open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists no symmetry at a given point $x\in D$ (so at all points by homogeneity). By a symmetry at $x$ I mean an element $s\in Hol(D)$ such that in a small neighborhood of $x$ only $x$ is fixed and $s^2=1$.

REPLY [14 votes]: E.Cartan proved in 1936 that for dimension 1 and 2 bounded homogeneous spaces are symmetric. For dimension 3 he did not publish the proof considering it loo long in comparison to the interest  of the result.  This has now changed with P-Sapiro's  example for dimension 4. So the proof for dimension 3 is presumably somewhere in E.Cartan's Nachlass, unpublished.