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Starting from a bundle τ : E → R , the bundle π : J 1 τ ∗ → E , which is the dual of the first jet bundle J 1 τ and a sub-bundle of T ∗ E , is the appropriate manifold for the geometric description of time-dependent Hamiltonian systems. Based on previous work, we recall properties of the complete lifts of a type ( 1 , 1 ) tensor R on E to both T ∗ E and J 1 τ ∗ . We discuss how an interplay between both lifted tensors leads to the identification of related distributions on both manifolds. The integrability of these distributions, a coordinate free condition, is shown to produce exactly Forbat’s conditions for separability of the time-dependent Hamilton–Jacobi equation in appropriate coordinates.