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I think quite a few names go well with Elizabeth but don't sound too matching in a lizzie Izzie way. Think you need something slightly more formal sounding and fairly timeless to go with Elizabeth.
btw just looked on a baby names site and it said that Isobel is Scottish in origin and means Consecrated to God whereas Elizabeth is Hebrew originally and means Oath of God. Not sure how accurate that is.
Don't think there's an issue. Although both Isobel (from Isabella) and Elizabeth are of the same origin: "Derived from the elements 'el' which means God ; 'sheba' which means oath" - they are often used today as independent names, with many variants of their own, eg. Isobel, Izabella, Isabeau, for Isabella, which are seldom classified as variants of Elizabeth. More info on the name Isabella, and the name Elizabeth.
Isabel is Elizabeth in Spanish (ie Reina Isabel is our monarch if you're reading Hola instead of Hello). I always think that if there's one small reason not to use a name, then don't. It's not as they're aren't lots to choose from.
Were you planning to use those shortenings? If not I can't see an issue with it. After all you could end up with Lizzy and Bella or Beth and Izzy or even use their whole names. But yes there are many other names you could choose from and avoid the whole issue
Oooh Sorry that would be silly. Everybody knows that Isabel is Spanish for Elizabeth. What about Elizabeth and Annabel? They match better imo. They are both steadier classics. Isabella more of the moment than Annabel.