I need to cover a case of $n$-dimensional locally symmetric Riemannian space of rank $n-1$. Is there a simple proof that there is no such irreducible space ($n>4$)? If I need to cite the Cartan classification for that, which concrete paper/book you suggest?



Riemannian symmetric space of dimension $n$ and rank $n-1$
An irreducible symmetric space $M$ of dimension $n$ and rank $n-1$ has dimension $n=2$.
Indeed, the rank is the codimension of a principal orbit of the isotropy group $K$ at $p \in M$ acting on the tangent space $T\_pM$. So the $K$-orbits are one dimensional. Now any monoparametric subgroup of $K$ has a one dimensional orbit contained in a $2$-dimensional subspace of $T\_pM$ hence $K$ has a one dimensional orbit contained in a $2$-dimensional subspace of $T\_pM$. So $K$ has an invariant $2$-dimensional subspace in $T\_pM$. But $K$ is the holonomy group of $M$ at $p$ hence $K$ acts irreducibly on $T\_pM$. Then $dim(T\_p M)=2$.


