A process for purifying an exhaust gas comprising carbon monoxide, hydrocarbons, nitrogen oxides, and sulfur oxides by periodically adjusting exhaust gas conditions from lean-burn to fuel-rich and by using a catalyst comprising an alumina support with lithium dissolved therein to form a solid solution, at least one first ingredient selected from the group consisting of platinum and palladium loaded on the alumina support, and at least one second ingredient selected from the group consisting of alkali elements, alkaline-earth elements, and rare-earth elements loaded on the support. Sulfates formed from sulfur oxides in the exhaust gas are decomposed by the lithium under the fuel-rich conditions to inhibit the alumina support from being acidified.