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While free/non-potential magnetic energy is a necessary element of any active phenomenon in the solar corona, its role as a marker of the trigger of eruptive process remains elusive. Based on the unique decomposition of the magnetic field into potential and non-potential components, magnetic energy and helicity can also both be uniquely decomposed into two quantities. Using two 3D MHD parametric simulations of a configuration that can produce coronal jets, we compare the dynamics of the magnetic energies and of the relative magnetic helicities. Both simulations share the same initial set-up and line-tied bottom-boundary driving profile. However, they differ by the duration of the forcing. In one simulation the system is driven sufficiently so that an helical jet is induced. The generation of the jet is however markedly delayed: a relatively long phase of lower-intensity reconnection takes place before the jet is eventually induced. In the other reference simulation, the system is driven during a shorter time, and no jet is produced. As expected, we observe that the Jet producing simulation contains a higher value of non-potential energy and non-potential helicity. Focusing on the phase between the end of the driving-phase and the jet generation, we note that magnetic energies remain relatively constant, while magnetic helicities have a noticeable evolution. During this post-driving phase, the ratio of the non-potential to total magnetic energy very slightly decreases while the helicity eruptivity index significantly increases. The jet is generated when the system is at the highest value of this helicity eruptivity index. This proxy critically decreases during the jet generation phase. The free energy also decreases but does not present any peak when the jet is being generated.