Complex sentence: Yareena challenges Yar to a fight to the death to claim back the position and while Picard objects to the fight, Lutan refuses to give the "Enterprise" the rest of the vaccine unless Yar participates.
Split into simpler sentences: Yareena challenges Yar to a fight to the death to claim back the position. When Picard objects to the fight, Lutan refuses to give the "Enterprise" the rest of the vaccine unless Yar participates.