The set of $n \times n$ symmetric matrices over $\mathbb R$ form a symmetric space. The relevant Lie group is $G = GL_n(\mathbb R)$ and the relevant involution is $\sigma(X)=X^{-T}$; it follows then that $G_{\sigma} = \{X \in G \mid \sigma(X)=X^{-1}\}$ is precisely the set of symmetric matrices. Notice also that $G^{\sigma} = \{X \in G \mid \sigma(X)=X\}$ is precisely the set of orthogonal matrices.
The symmetric matrices satisfy the spectral theorem. I think that the symmetric matrices constitute merely one example of a symmetric space.
I believe that whenever $\sigma$ is a Cartan involution, an analogue of the spectral theorem will exist. I think this should say that there exists a maximal abelian subgroup $A \subset G_{\sigma}$ such that for all $x \in G_{\sigma}$ there exists a $k \in G^{\sigma}$ and a $d \in A$ such that $x = kdk^{-1}$. Is there a reference for this?
Is there some vague generalisation of the spectral theorem for all symmetric spaces? This should say that something about the set $\{ k x k^{-1} \mid k \in G^{\sigma}\}$ given any $x \in G_{\sigma}$.
[Removed example of 2 so as to not confused the reader]
