Finely shred the red cabbage using a sharp knife or a mandolin.
Place them in a bowl filled with cold water and give them a quick wash, gently squeezing for a minute. Let them drain in a colander while preparing the brine.
In a bowl, combine ½ cup of white wine vinegar or apple cider vinegar, ¼ cup of lemon juice, ½ cup of room temperature water, 3 teaspoon of salt, and 2 teaspoon of sugar. Mix well until the salt and sugar are fully dissolved.
Place the shredded red cabbage in a clean glass jar, pressing it down firmly as you go. Pour the vinegar mixture over the cabbage, then press it down again with the back of a spoon so the brine reaches as much of the cabbage as possible. The cabbage may not be fully submerged at first, but it will soften and release liquid as it sits. If you prefer the cabbage completely covered straight away, make an extra half batch of brine.
Seal the jar tightly and store it in the refrigerator for up to 3 weeks. Your pickled red cabbage is ready to it in only two hours.