For the sodium alginate bath: Combine the water, sugar, and sodium alginate in a pot and bring to a boil.
Remove the pot from the heat, and allow the mixture to cool to room temperature. Store, refrigerated, overnight. Discard this mixture after 10 days of use.
For the lemon soda base: Combine the lemonade and the sugar in a narrow bain-marie. Pour in the calcium lactate and shear with a handheld blender.
Continue to shear and add the xanthan gum, pouring it in slowly. Shear for 45 seconds to 1 minute.
Pour this liquid mixture into the cavities of a 1.5-cm/.75-in diameter silicone dome mold and freeze. Meanwhile, warm up the sodium alginate bath to 80°C/175°F. Warm a large water bath for rinsing the spheres to the same temperature, and warm the simple syrup to hold the spheres.
Once the domes are frozen hard, remove them from the mold and place them inside the hot sodium alginate bath for 1 minute. They will still be frozen at the core.
Take the spheres out of the sodium alginate bath using a slotted spoon and rinse each one in the hot water. Place the spheres in the hot simple syrup. Allow it to cool before refrigerating. Discard after 36 hours.
Place the spheres and the lemonade inside a soda siphon. Alternatively, you can use a cream siphon (see the sidebar). The charges for soda siphon and cream siphons are interchangeable between both apparatus, but for carbonation for the soda siphon, you will need CO2 (cream uses N2O chargers).
Close the lid on the siphon tightly and load two charges if using a 1-liter siphon. There are no smaller versions of the soda siphon; there are, however, smaller versions of the cream siphon, in which case you will need only one charge. Let it sit refrigerated overnight. The CO2 needs to permeate the spheres.
Just before service, release the pressure from the siphon before opening the lid. Transfer the carbonated spheres to a vacuum-seal canister (see Resources) and vacuum seal it. This will preserve the carbonation in the spheres during service. You will have to do this (vacuum seal) after each dessert is served; otherwise the carbonation in the sphere will decrease. Reserve the spheres in multiple canisters (about 10 spheres per canister). This canister can also be used with the more economical vacuum sealers; the canister comes with a tube attachment that can be connected to the sealer to extract the air from the canister. Discard the carbonated spheres after service. No matter how well it is sealed, it will lose its carbonation by the next day.