I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all eigenvalues of Ricci curvature ${\rm Ricci}(M)$ strictly positive and why does each eigenvalue have a multiplicity of at least 2?
Could you please give me some help with the details? Thanks in advance. By the way, are there some nice references for these questions to refer to?
We first assume that we have an irreducible symmetric space $G/K$. The adjoint action of $K$ on $\it p$ (the orthogonal of the lie algebra of $K$) is irreducible, and therefore admits a unique (up yo homothety) invariant quadratic form. Therefore, there exist a constant $c$ such that the Ricci quadratic form (computed at some point) is $c$ time the Killing form at this point. But these two tensors are invariant so globally $R= c\times g$.
Then if we start with something not irreducible, up to a finite cover, the manifold is a product of compact irreducible symmetric spaces and a flat torus.