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 $s^2=1$.
There are some examples given by PjateckiĭŠapiro in Classification of bounded homogeneous regions in ndimensional complex space. Dokl. Akad. Nauk SSSR 141 1961 316–319. and On bounded homogeneous domains in an ndimensional complex space. Izv. Akad. Nauk SSSR Ser. Mat. 26 1962 107–124. (I have a vague memory that the smallest examples are 4dimensional, but might have misremembered.) 


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 PSapiro's example for dimension 4. So the proof for dimension 3 is presumably somewhere in E.Cartan's Nachlass, unpublished. 

