It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example Chapter V Theorem 8.1 in Brocker). In other words, the universal covering map $\tilde{G}\times \mathbb{R}^n\to G$ factors through some $\tilde{G}\times\mathbb{T}^n$.

More generally, consider a compact globally symmetric space $M$. Then the universal cover of $M$ is the product of a simply connected globally symmetric space $\tilde{M}$ of compact type and a Euclidean space $\mathbb{R}^n$. My question is, is it still true that the universal covering map $\tilde{M}\times \mathbb{R}^n\to M$ factors through some $\tilde{M}\times \mathbb{T}^n$ so that $\tilde{M}\times \mathbb{T}^n\to M$ is a finite Riemannian cover? Any help is appreciated. Thank you!

isometricsplitting $\tilde{M}\times \mathbb{T}^n$? $\endgroup$