The invention relates to a process for workup of a stream (1) comprising butene and/or butadiene, butane, hydrogen and/or nitrogen and carbon dioxide, comprising: (a) absorption of stream (1) with a mixture (5) comprising 80 to 97% by weight of N-methylpyrrolidone and 3 to 20% by weight of water to obtain a stream (9) comprising N-methylpyrrolidone, water, butene and/or butadiene, butane, and optionally carbon dioxide, and a stream (7) comprising hydrogen and/or nitrogen and butane, (b) extractive distillation of stream (9) with a stream (13) comprising 80 to 97% by weight of N-methylpyrrolidone and 3 to 20% by weight of water to separate the stream (9) into a stream (17) comprising N-methylpyrrolidone, water, butene and/or butadiene, and a stream (15) comprising essentially butane, and optionally carbon dioxide, (c) distillation of stream (17) into a stream (23) comprising essentially N-methylpyrrolidone and water, and a stream (21) comprising butene and/or butadiene.