Bring all ingredients to a boil in a small, heavy-bottomed saucepan, stirring constantly and mashing strawberries a bit with the back of a spoon.

Cook, still stirring, for two to three minutes until thickened (mixture will be slightly chunky). Remove from heat.

Combine the first five ingredients in a small, heavy-bottomed saucepan. Stir over medium heat until slightly thickened.

Add water and stir until thickened, about five minutes. Add oil and stir. Remove from heat.

Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius). Grease two nine x 13-inch (20×30-centimeter) baking pans and set aside.

Combine eggs and sugars in a large mixing bowl. Beat on high speed until pale and thick, about five minutes. Add oil and beat on medium speed; add remaining ingredients and mix until incorporated.

Divide batter between the prepared pans. Drop tablespoonfuls of strawberry filling in even intervals over the batter in one pan. Repeat with the lemon filling over the batter in the second pan.

Bake for 50 minutes, or until cake begins to brown at edges and a toothpick inserted comes out clean. Cut into squares.