It is argued that the integrable model for the planar spectrum of the AdS/CFT correspondence can accommodate for the full spectrum of excitations D^αα˙,ϕ^[IJ],ψ^I,ψ¯_I,F^αβ,F˜^α˙β˙ (with I,J∈1…4) if double excitations are allowed for all three lowering operators of the internal SU(4) group as well as the supersymmetries. We present a tree-level analysis of related creation amplitudes in the nested Bethe ansatz as well as in the original level-1 picture in which excitations of various flavours scatter by a true S-matrix. In the latter case, the creation amplitudes for all double excitations we encounter take a perfectly universal form. Building on these ideas we work out Bethe solutions and states relevant in the mixing problem concerning the on-shell Lagrangian of N=4 super Yang-Mills theory. Owing to the very existence of double excitations, the chiral Yang-Mills field strength tensor can be represented by the four fermions {ψ³¹,ψ³²,ψ⁴¹,ψ⁴²} moving on a spin chain of length two. Our analysis remains restricted to leading order in the coupling, where the conformal eigenstate corresponding to the on-shell Lagrangian only comprises the pure Yang-Mills action. It should eventually be possible to augment our analysis to higher loop orders by incorporating coupling corrections in the relevant ingredients from the Bethe ansatz. Finally, it was recently realised how structure constants for operators containing the hitherto hidden half of the excitations can be computed by the hexagon formalism. We use this for a first test of our conjecture for the on-shell Lagrangian, namely that its three-point function with two half-BPS operators of equal length ought to vanish.