Thanks Donu for the quick answer. So could you give me more details on how you get the $g^2$?

Right, good point!

@Ryan, yes I meant that the complement of the unknot in $\mathbf{R}^3$ deformation retracts to $S^2\vee S^1$.

You are right completely right Ryan, I forgot the point at $\infty$, it is the unknot complement in $\mathbf{R}^3$ which is homotopy equivalent to a countable wedge of $S^2$. I'll remove my motivation.

Thanks Prof. Helgason for the short genealogy of the problem. By the way, I discovered a nice paper of Borel: Les fonctions automorphes de plusieurs variables complexes, where on p. 177 he mentions Cartan's result in the case n=3 and points out that the problem was still open (in 1952) for n=4.

Thanks a lot Ralph for the reference on Cohen Macaulay rings.

It is possible though to save my idea about using unicity of $\rho_v^F$ by saying that is is the unique $F$-isometrie such that $s^2=1$, $s(w)=-w$ for all $w\in\mathbf{C}v:=L$ and such for all $w$ which are $F$-perp to $L$ one has $s(w)=w$.