Aioli is an emulsified sauce, meaning that liquids that don't usually combine -- such as oil and water or, in this case, oil, lemon juice, and vinegar -- are blended together.
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This recipe for aioli contains raw egg, which should not be used in food prepared for pregnant women, babies, young children, the elderly, or anyone whose health is compromised.
The key to preventing the sauce from separating is to have all the ingredients at room temperature. Be sure to use the freshest eggs and the plumpest, least blemished garlic you can find.