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I find NO reference of Obama Jr using the spelling "Barak" until AFTER his trip to Kenya in the early
80s, which includes his first discoveries of his father, Barak Sr, pictured here in 1959 before meeting Ann:

Of course, we have a different name used by Obama Jr after his mother married Lolo Soetoro, circa 1966:

Which differs yet again after Dec 1971, when his mom dumped him on the grandparents in Hawaii:

Obama Jr seems to have used "Barak", after visiting Kenya when meeting Barak Sr on his discovery tour, nearly only with black audiences.
He appears to have DROPPED the usage of that spelling when his presidential campaign started rolling in 2005. An FOIA check (or diligence by a group of Freepers) might determine if he used "Barak" on legal forms (drivers licenses, state income tax, vehicle registration, etc) while in:
-- California (Occidental)
-- New York (Columbia)
-- Massachusetts (Harvard)
-- Illinois (before running for office first in 1996)