space, and the Killing fields show how to "slide along" that submanifold. They span the tangent space of that submanifold. The tangent space should have the same dimension as the isometries acting effectively at that point. That is, one expects Yet, in general, the number of Killing fields is larger than the dimension of that tangent space. How can this be? The answer is that the "extra" Killing fields are redundant. Taken all together, the fields provide an over-complete basis for the tangent space at any particular selected point; linear combinations can be made to vanish at that particular point. This was seen in the example of the Killing fields on a 2-sphere: there are 3 Killing fields; at any given point, two span the tangent space at that point, and the third one is a linear combination of the other two. Picking any two defines the remaining degenerate linear combinations define an orthogonal space Cartan involution The Cartan involution is defined as the mirroring or reversal of the direction of a geodesic. Its differential flips the direction of the tangents to a geodesic. It is a linear operator of norm one; it has two invariant subspaces, of eigenvalue +1 and −1. These two subspaces correspond to and respectively. This can be made more precise. Fixing a point consider a geodesic passing through , with The involution is defined as This map is an involution, in that When restricted to geodesics along the Killing fields, it is also clearly an isometry. It is uniquely defined. Let be the group of isometries generated by the Killing fields. The function defined by is a homomorphism of . Its infinitesimal is The Cartan involution is a Lie algebra homomorphism, in that for all The subspace has odd parity under the Cartan involution, while has even parity. That is, denoting the Cartan involution at point as one has and where is the identity map. From this, it follows that the subspace is a Lie subalgebra of , in that As these are even and odd parity subspaces, the Lie brackets split, so that and The above decomposition holds at all points for a symmetric space ; proofs can be found in Jost. They also hold in more general settings, but not necessarily at all points of the manifold. For the special case of a symmetric space, one explicitly has that that is, the Killing