Preheat oven to 350°F. Spray an 8x8 baking pan with non stick baking spray. Set aside.
In a large bowl combine the cake flour, baking powder, baking soda, and 3/4 cup of white sugar. Stir together with a fork to mix.
In another medium size mixing bowl combine the melted butter, buttermilk, rum, oil, vanilla extract and egg yolks. Stir to combine.
In a small separate bowl add the 2 egg whites. (you can discard or keep the other 2 remaining egg whites for another recipe, or an egg white omelet.) Beat the egg whites with a hand mixer until soft peaks start to form. Add the remaining 1/4 cup white sugar and continue to beat on high until peaks are stiff and shiny.
Add the wet ingredients to the dry, stir until combined. Then fold in the egg whites gently.
Pour the batter into the prepared cake pan.
Bake at 350 for about 25-28 minutes or until the cake is cooked through. A toothpick inserted in the cake will come out clean, no batter.