This recipe for homemade ginger cordial, courtesy of Allen Katz, is used in his delicious cocktail, Eli's Elixir.
Slice the top and bottom of lemon with a paring knife. Set one of the flat sides on your work surface. From top to bottom, following the curve of the fruit, cut away peel. Trim off any white pith from peel and discard. Discard flesh or save for another use.
In a medium saucepan, bring sugar, 1 cup water, ginger, cream of tartar, and lemon peel to a boil over high heat, stirring occasionally, until sugar has dissolved, about 5 minutes.
Add lemon juice and return to a boil. Strain mixture through a fine mesh sieve into a clean bottle; let cool. Keep refrigerated, until ready to use, up to 2 weeks.