  We consider the near-horizon geometry of supersymmetric extremal black holes
in un-gauaged and gauged 5-dimensional supergravity, coupled to abelian vector
multiplets. By analyzing the global properties of the Killing spinors, we prove
that the near-horizon geometries undergo a supersymmetry enhancement. This
follows from a set of generalized Lichnerowicz-type theorems we establish,
together with an index theory argument. As a consequence, these solutions
always admit a $\mathfrak{sl}(2,\mathbb{R})$ symmetry group.
