Given an orientable Riemannian manifold, we consider the bundle of oriented orthonormal frames and the tangent sphere bundle over it, which admit natural Riemannian metrics defined by the Riemannian connection. We show that there is a natural homomorphism between the Lie algebras of fiber preserving Killing vector fields on these bundles. In particular, for any orientable Riemannian manifold of dimension two, we show that the homomorphism is extended to an isomorphism between these Lie algebras.
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