Low-emission electrolytic hydrogen (LEH), produced electrolytically using renewable energy, is essential for decarbonising the global economy, particularly that of the European Union. However, high-temperature electrolysis processes need a Hydrogen Emissions Intensity Index to accurately assess emissions intensity during hydrogen production. That index must incorporate the thermodynamic performance of the electrolysis solution and associated plants for energy and mass balance (BoP) because of the different mature technologies (Technology Readiness Level >8): Alkaline Electrolysis, Proton Exchange Membrane, or Solid Oxide Electrolyte. This study establishes, for the first time, as the author's knowledge, a Hydrogen Emissions Intensity Index (IEH2) for high-temperature electrolysis based on Solid Oxide Electrolyte (SOE), in which heat demand is supplied by an external thermal subsystem coupled employing a Balance of Plant. For this, all the energy is provided by renewable sources, i.e. PV and Biomass. The findings reveal the strong dependence of IEH2 on the SOE cells' operating voltage and the associated BoP and thermal subsystems' key parameters (thermal efficiency and efficacy). The solid membrane or electrolyte thickness appears to significantly influence operational carbon emissions at the cost of degrading the electrolyser's thermodynamic performance. The study also explores how the BoP and thermal subsystems' performance impact IEH2, establishing a trade-off with the thermodynamic performance of the SOE stack. This Hydrogen Emissions Intensity Index is proposed as a valuable tool for optimising future LEH production for SOE and other electrolysis techniques by including emissions intensity as a key parameter.