The proper name and place name is spelt just as you have in the question - only with a capital initial letter - Lincoln.
Abraham Lincoln
Lincoln
Lincoln. It was named for Abraham Lincoln.
It is spelled Abraham Lincoln as in the question.
As in Abraham? Lincoln
Illinois
Gordon Lincoln ARVIDSON has written: 'Learning to spell'
LINCOLN
Abraham Lincoln
Abraham Lincoln.
The spelling is "Abe Lincoln", a short name form for President Abraham Lincoln.
As Abraham Lincoln is a person's name you would write it as you see it in English.