There is great current interest in multicomponent superhydrides due to their unique quantum properties under pressure. A remarkable example is the ternary superhydride Li_{2}MgH_{16} computationally identified to have an unprecedented high superconducting critical temperature T_{c} of ∼470 K at 250 GPa. However, the very high synthesis pressures required remains a significant hurdle for detailed study and potential applications. In this Letter, we evaluate the feasibility of synthesizing ternary Li-Mg superhydrides by the recently proposed pressure-potential (P^{2}) method that uniquely combines electrochemistry and applied pressure to control synthesis and stability. The results indicate that it is possible to synthesize Li-Mg superhydrides at modest pressures by applying suitable electrode potentials. Using pressure alone, no Li-Mg ternary hydrides are predicted to be thermodynamically stable, but in the presence of electrode potentials, both Li_{2}MgH_{16} and Li_{4}MgH_{24} can be stabilized at modest pressures. Three polymorphs are predicted as ground states of Li_{2}MgH_{16} below 300 GPa, with transitions at 33 and 160 GPa. The highest pressure phase is superconducting, while the two at lower pressures are not. Our findings point out the potentially important role of the P^{2} method in controlling phase stability of complex multicomponent superhydrides.