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Eddie Izzard aims to stand in Brighton at next general election
(www.bbc.co.uk)
Welcome Brightonians and visitors alike!
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If she completely rebranded to Suzy then a lot of people wouldn't know who you were talking about, and she'd lose so much public affiliation. As you say, Eddie Izzard is a stage name as much as it is a brand name.
Eddie Izzard in public, Suzy in private. Makes sense.