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It is known that a geodesic Y in an abstract reflection space X in the senseof Loos, without any assumption of differential structure, canonically admitsan action of a 1-parameter subgroup of the group of transvections of X. In thisarticle, we prove an analog of this result stating that, if X contains anembedded hyperbolic plane H, then this yields a canonical action of a subgroupof the transvection group of X isomorphic to a perfect central extension ofPSL(2,R). This result can be further extended to arbitrary Riemannian symmetricspaces of non-compact type embedded in X and can be used to prove that aRiemannian symmetric space and, more generally, the Kac-Moody symmetric spaceG/K for an algebraically simply connected two-spherical Kac-Moody group Gsatisfies a universal property similar to the universal property that the groupG satisfies itself.

An extension theorem for embedded Riemannian symmetric spaces of non-compact type and an application to their universal property