Let $G$ be a semisimple Lie group and let $G = KAK$ be a Cartan decomposition.


For $\mathrm{SL}\_2(\mathbb{R})$ it holds for every $g \in G$ that $KgK = Kg^{-1}K$.


Does the same hold for every semisimple Lie group?



Question on KAK decomposition
No, this fails already in $\mathrm{SL}\_3(\mathbb{R})$. If $Kg\_1K=Kg\_2K$, then $g\_1^Tg\_1$ is conjugate (by an element of $K$) to $g\_2^Tg\_2$. So $g\_1^Tg\_1$ and $g\_2^Tg\_2$ have the same (positive eigenvalues). In particular, if $g\_1$ and $g\_2$ are positive diagonal matrices, then they have the same diagonal entries up to permutation. This rarely happens when $g\_1$ and $g\_2$ are inverses of each other. For example, $\mathrm{diag(2,2,1/4)}$ and $\mathrm{diag(1/2,1/2,4)}$ have very different diagonal entries, and they are inverses of each other.


