Carbon cannot be used in the reduction of ${\mathrm{Al}}_2{\mathrm{O}}_3$ because :

Carbon cannot be used in the reduction of ${\mathrm{Al}}_2{\mathrm{O}}_3$ because the heat of the formation of ${\mathrm{Al}}_2{\mathrm{O}}_3$ is too high

The enthalpy of formation is extremely high for alumina.

For the carbothermic reduction to occur, extremely high temperatures of roughly 2100 degrees Celsius are needed.

Due to its high stability and location below carbon's oxidation line in the Ellingham Diagram, aluminium oxide ${\mathrm{Al}}_2{\mathrm{O}}_3$ cannot be reduced by carbon due to unfavourable energy conditions.