Molten carbonate electrolysis cells represent a key technology for harnessing surplus energy from renewable sources and converting it into gaseous energy carriers. To optimize their efficiency, a comprehensive understanding of each step in the operational process is essential. Here, we focus on the electrolyte of choice in molten carbonate cells: the Li1.24K0.76CO3 melt. Utilizing molecular dynamics with explicit polarization, we demonstrate that the structure of this molten mixture is characterized by a dense network of lithium-carbonate complexes, with K+ ions loosely embedded within this network. This structural insight enables us to rationalize from an atomistic perspective the conductivity trends observed experimentally in molten carbonates. Moreover, our work highlights the importance of including polarization for the simulations of dense liquid carbonates. It also acts as a foundational step towards more advanced theoretical studies for elucidating the role of the electrolyte in these devices.