Cut 1 lemon and 1 lime in half, and combine the juice with 1 to 1 ½ ounces of simple syrup and 6 to 8 ounces of water. I use a boston shaker because it’s easy to load and clean, but you could use any water tight container.
Once the other ingredients are in the shaker, I load 6 large ice cubes in as well. This allows the drink to be enjoyed immediately, although it does also introduce more water. So, if you choose not to shake with ice, compensate with a few extra ounces.
With the shaker prepared, I shake lengthwise to fully incorporate the ingredients for about 30 seconds. At this point it should be fairly cold to the touch. Finally I strain the drink as I pour it, using both a hawthorn strainer and a conical fine mesh sieve. This fully filters any ice, pulp, or seeds while not restricting flow. I then add some more ice cubes to the cup and enjoy.
For something more akin to an ultra-fresh sprite, you can carbonate your juice. This assumes your machine allows you to carbonate more than just water, which as far as I am aware is just the Drinkmate I have or a Brevil. I find just substituting carbonated water after shaking the other ingredients isn’t concentrated enough to make a noticeable difference. But I normally skip this entirely.