## Abstract

By [5] it is known that a geodesic *Îł* in an abstract reflection space *X* (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of *X*. In this article, we modify these arguments in order to prove an analog of this result stating that, if *X* contains an embedded hyperbolic plane đť“— âŠ‚ *X*, then this yields a canonical action of a subgroup of the transvection group of *X* isomorphic to a perfect central extension of PSL_{2}(â„ť). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type *Y* lying in *X* and can be used to prove that a Riemannian symmetric space and, more generally, the Kacâ€“Moody symmetric space *G/K* for an algebraically simply connected two-spherical split Kacâ€“Moody group *G*, as defined in [5], satisfies a universal property similar to the universal property that the group *G* satisfies itself.

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