I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely).

The first big step which I find difficult to break to concise little pieces is how to get from general riemannian locally symmetric spaces to a product of simply connected indecomposable symmetric spaces. Here's my progress so far:

Let $(M,g,\nabla)$ be a riemannian manifold with a levi-civita connection (of dimension $n$).

For every point $p \in M$ the linear reflection on the tangent space $-Id: T_p M \to T_p M$ extends to define an involution on every normal coordinate ball $\sigma_p : N_p \to N_p$ which corresponds to reversing the geodesics through $p$.

Definition:A locally symmetric space is a riemannian manifold $(M,g)$ with the property that for every $p \in M$ the geodesic involution $\sigma_p :N_p \to N_p$ is an isometry (or equivalently affine with respect to the connection $\nabla$).

Here's how I understand the steps so far:

**$M$ and $\nabla$ are analytic**- Any manifold with a connection $\nabla$ for which both torsion and curvature are parallel is analytic and $\nabla$ is analytic. Kobayashi & Nomizu VI 7.7**Assume $M$ is complete**- As far as I understand we can't make progress without this.**Universal cover**- The universal cover $\pi : \tilde M \to M$ inherits a metric structure for which it is an isometry implying $\tilde M$ is again locally symmetric. We assume from now on that $M$ is simply connected.**Involutions extend to global isometries**(?) - Kobayashi and Nomizu IV 6.3 - "Let $M \supset U \to N$ be an isometric immersion of connected analytic riemannian manifolds with $M$ complete and $N$ simply connected then $f$ extends to an isometric immersion $f: M \to N$". So the isometry group $I(M)$ acts transitively and $M = I(M)/K$ where $K$ is a compact stabilizer group. Furthermore $K \subset O(T_p M)$ at every point.**Holonomy is subgroup of stabilizer**- Transvections $T^{\gamma}_t := \sigma_{\gamma(t/2)} \circ \sigma_{\gamma(0)}$ along geodesics $\gamma : I \to M$ form one parameter families of isometries (fixing correspondingly the points $\gamma(0)$). Every curve is $C^1$ limit of piecewise geodesic curves. Let $c : I \to M$ be a closed curve which is a limit of closed polygon geodesics. Transvections must fix the edges of the polygons and therefore define in the limit a parallel translation. $K$ Is compact in $I(M)$ in $I(M)$ (being a stabilizer subgroup). Therefore $Hol_p \subset K$.**De Rahm decomposition theorem**(?) - There's a question about this on the site but it hasn't received very useful answers. I'd like to know where to find the statement and the proof of this theorem in the most modern language.

Adding everything together we have a product decomposition $I(M_0)/H_0 \times \dots \times I(M_k)/H_k$ where $I(M_j)$ acts irreducibly. From here on it's pretty much an algebraic road I think.

Is there something substantial missing/wrong in the above outline?