  This article concerns cotangent-lifted Lie group actions; our goal is to find
local and ``semi-global'' normal forms for these and associated structures. Our
main result is a constructive cotangent bundle slice theorem that extends the
Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies
to all proper cotangent-lifted actions, around points with fully-isotropic
momentum values. We also present a ``tangent-level'' commuting reduction result
and use it to characterise the symplectic normal space of any cotangent-lifted
action. In two special cases, we arrive at splittings of the symplectic normal
space, which lead to refinements of the reconstruction equations (bundle
equations) for a Hamiltonian vector field. We also note local normal forms for
symplectic reduced spaces of cotangent bundles.
