Preheat oven to 350 F. Grease an 8" square pan and set aside.
In a large bowl, cream with butter, oil, and sugar until fluffy. Add the eggs one at a time, beating between each addition.
Add the lemon zest, lemon juice, rose water, and vanilla. Beat well.
Add the flour, cardamom, baking powder, and salt. Beat just until mixed.
Scrape the batter into the prepared pan. Bake for 25-30 minutes or until a wooden skewer inserted into the centre comes out clean. Remove from oven and let cool on a wire rack.