We investigate the relaxation mechanism of a supercooled tetrahedral liquid at its limit of stability using isothermal isobaric ($NPT$) Monte Carlo (MC) simulations. In similarity with systems which are far from equilibrium but near the onset of jamming [OHern et.al., Phys. Rev. Lett. {bf 93}, 165702 (2004)], we find that the relaxation is characterized by two time-scales: the decay of long-wavelength (slow) fluctuations of potential energy is controlled by the the slope $[partial (G/N)/partial phi]$ of the Gibbs free energy ($G$) at a unique value of per particle potential energy $phi = phi_{mid}$. The short-wavelength (fast) fluctuations are controlled by the bath temperature $T$. The relaxation of the supercooled liquid is initiated with a dynamical crossover after which the potential energy fluctuations are biased towards values progressively lesser than $phi_{mid}$. The dynamical crossover leads to the change of time-scale, i.e., the decay of long-wavelength potential energy fluctuations (intermediate relaxation). Because of the condition [$partial^2 (G/N)/partial phi^2 = 0$] at $phi = phi_{mid}$, the slope $[partial (G/N)/partial phi]$ has a unique value and governs the intermediate stage of relaxation, which ends just after the crossover. In the subsequent stage, there is a relatively rapid crystallization due to lack of long-wavelength fluctuations and the instability at $phi_{mid}$, i.e., the condition that $G$ decreases as configurations with potential energies lower than $phi_{mid}$ are accessed. The dynamical crossover point and the associated change in the time-scale of fluctuations is found to be consistent with the previous studies.
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