In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a subspace $V$ of $T\_{p}Q$ such that $[u, [v, w]] \in V$ for all $u,v,w \in V$; see for example Theorem 7.2 in Helgason, *Differential geometry, Lie groups, and symmetric spaces*.


As far as I understand, the proof carries over directly to the case where the ambient manifold is a *semi-Riemannian* symmetric space.


**Question**. Does anyone know where in the literature I can find a statement/proof of this more general result?



Proof of equivalence between Lie triple systems and totally geodesic submanifolds
Try pages 71 and 72 of Cartan's "La géométrie des groupes de transformations" <https://eudml.org/doc/235668>


