Complex sentence: Lieberman, despite intense negotiations in search of a compromise by Reid, refused to support a public option; a concession granted only after Lieberman agreed to commit to voting for the bill if the provision was not included, even though it had majority support in Congress.
Split into simpler sentences: Lieberman, despite intense negotiations in search of a compromise by Reid, refused to support a public option; a concession granted only after Lieberman agreed to commit to voting for the bill if the provision was removed. To win filibuster-proof support for the overall bill, the provision was not included, even though it had majority support in Congress.