  Lorentz invariant derivative interactions for a single spin-2 field are
investigated, up to the cubic order. We start from the most general Lorentz
invariant terms involving two spacetime derivatives, which are polynomials in
the spin-2 field as well as its first derivatives. Using a perturbative
Arnowitt-Deser-Misner analysis, we determined the parameters such that the
corresponding Hamiltonian possesses a Lagrange multiplier, which would signify
there are at most 5 degrees of freedom that are propagating. The resulting
derivative terms are linear combinations of terms coming from the expansion of
the Einstein-Hilbert Lagrangian around a Minkowski background, as well as the
cubic "pseudolinear derivative term" identified in Hinterbichler [J. High
Energy Phys. 10 (\textbf{2013}) 102]. We also derived the compatible potential
terms, which are linear combinations of the expansions of the first two de
Rham-Gabadadze-Tolley mass terms in unitary gauge.
