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

In Ellingham diagram, the ${\mathrm{\Delta }}_{\mathrm{f}}{\mathrm{G}}^{\mathrm{o}}\text{\hspace{0.17em}of\hspace{0.17em}}{\mathrm{Al}}_2{\mathrm{O}}_3$ lies below that of ${\mathrm{CO}}_2$. If reduction is carried out at very high temperature, the Al produced will react with carbon forming ${\mathrm{Al}}_4{\mathrm{C}}_3$.