How? Under what law did Obama lose his US citizenship by going to Indonesia?
Hague Convention.
US does allow parents to relinquish their children’s citizenship and when it is done it IS recognized by the US.
It is true that a child does not lose their RIGHT to reclaim this citizenship if they so choice when they reach majority. Indonesian law did not recogize dual citizens. When SAD marries Lolo Soetoro AND moves to Indonesia the Indonesian government considers her and her child citizens of Indonesia. With the 1965 marriage when Obama was only 3 years old and the action of removing Obama from her US passport in 1968 when he was only 6 or 7 SAD is admitting - he is not a US citizen.
She had to do that for his own safety in Indonesia. It was done in 1968 - right after her and Obama II moved to Indonesia.
Indonesian law allows for a child to renounce their citizenship when the child is adopted. One of those laws is when a natural father reclaims their natural son. And that is what happened in December 1971.
Getting back to 1961 in the records is likely difficult. It is likely difficult for the Hawaiian officials. Assuming Obama changed identity/guardianship/citizenship at various times the records are likely seal, spun and mutilated. The wormhole of 1971 probably results in sealed records - meaning no one has really seen the ‘original’ BC for a long, long time.