When compared to ${\mathrm{\Delta G}}^{\mathrm{o}}$ for the formation of ${\mathrm{Al}}_2{\mathrm{O}}_3$, the ${\mathrm{\Delta G}}^{\mathrm{o}}$ for the formation of ${\mathrm{Cr}}_2{\mathrm{O}}_3$ is 

When compared to $∆{\mathrm{G}}^0$ for the formation of ${\mathrm{Al}}_2{\mathrm{O}}_3$​, the $∆{\mathrm{G}}^0$ for the formation of ${\mathrm{Cr}}_2{\mathrm{O}}_3$​ is lower (less negative). Hence, aluminum is used as reducing agent for reduction of ${\mathrm{Cr}}_2{\mathrm{O}}_3$​.
${\mathrm{Cr}}_2{\mathrm{O}}_3 + 2\mathrm{Al} \to {\mathrm{Al}}_2{\mathrm{O}}_3 + 2\mathrm{Cr}$
The reaction proceeds in the forward direction as ${\mathrm{Al}}_2{\mathrm{O}}_3$​ is more stable than ${\mathrm{Cr}}_2{\mathrm{O}}_3$​.