  In this article, relations between the root space decomposition of a
Riemannian symmetric space of compact type and the root space decompositions of
its totally geodesic submanifolds (symmetric subspaces) are described. These
relations provide an approach to the classification of totally geodesic
submanifolds in Riemannian symmetric spaces. In this way a classification of
the totally geodesic submanifolds in the complex quadric $Q^m :=
\SO(m+2)/(\SO(2) \times \SO(m))$ is obtained. It turns out that the earlier
classification of totally geodesic submanifolds of $Q^m$ by Chen and Nagano is
incomplete: in particular a type of submanifolds which are isometric to
2-spheres of radius $\tfrac{1}{2}\sqrt{10}$, and which are neither complex nor
totally real in $Q^m$, is missing.
