Let $Z$ be a compact, connected, orientable (**Edit: as Misha point out**) and locally Riemannian symmetric space. As a complete, simple connected, locally symmetric space is a global symmetric space. We can write $Z=\Gamma \backslash G/K$. One of such $(G,K)$ is $G=\mathrm{Iso}(\widetilde{Z})$,$K$ is the stablizer of a point in $\widetilde{Z}$.

My question is, if $\widetilde{Z}$ is of non compact type, can we always choose $(G,K,\Gamma)$ such that

- $G,K$ is connected.
- $\Gamma\subset G$ closed discrete?

If not, under what natural condition, $Z$ is this type.

One of naive choice is to take $G=\mathrm{Iso}^0(\widetilde{Z})$, but it is to always true that $\Gamma\subset \mathrm{Iso}^0(\widetilde{Z})$.

**Edit:**
I pose this question because in papers of H. Moscovici and R.J. Stanton
http://link.springer.com/article/10.1007%2FBF01393895
http://link.springer.com/article/10.1007%2FBF01232263
where proved the result for this type of space(maybe I missed something) but state the theorem for locally symmetric space.