# Maximal symmetry and isometries not connected to the identity

A pseudo-Riemannian manifold $M$ of dimension $n$ is said to be maximally symmetric if the space of its Killing vector fields has $n(n+1)/2$ dimensions.

If $M$ is maximally symmetric, then we have the following: for every $p\in M$ and every linear isometry $\Lambda:T_pM\to T_pM$ connected with the identity, there exists an isometry $\sigma:M\to M$ such that $\sigma(p)=p$ and $d\sigma_p=\Lambda$.

On the other hand, all maximally symmetric spaces I know of (flat spaces, sphere, hyperbolic space, de Sitter and Anti de Sitter spacetimes) have a stronger property: for every $p\in M$ and every linear isometry $\Lambda:T_pM\to T_pM$ (not necessarily connected with the identity), there exists an isometry $\sigma:M\to M$ such that $\sigma(p)=p$ and $d\sigma_p=\Lambda$.

My question is: is this stronger property a consequence of maximal symmetry? In case it is not, I would like to know of an example of a maximally symmetric space not having this stronger property.

Careful: the existence of $n(n+1)/2$ dimensional space of Killing fields does not imply global homogeneity. Think of any open subset of Euclidean space; typically they have no global isometries.