It was to prove to be the event of Iceland Airwaves, a curtain closing on the existence of one of the 21st century's most vital experimentalists. As The Knife took to the stage in Reykjavík, just past 10pm, the audience celebrated as if a victory was taking place – and in many ways it was, irrespective of the fact that everyone there knew it was to be the last time.

Karin Dreijer Andersson and Olof Dreijer invited debate and divide when they dropped Shaking the Habitual in 2013. With its aggressive, alien rhythms and harrowing synth lines, it was a fearless and fascinating statement – engulfed by an anti-everything atmosphere that makes something so sophisticated and ambitious feel like an underdog. Both refer to the self-released album as a resolutely political record, a necessary departure from earlier work like Deep Cuts and Silent Shout to escape the clutches of stagnation. 

So nobody expected the duo to announce their split in September, when they told Dazed that the Shaking the Habitual run would be their last. "We will close down, it’s our last tour," they said in an interview. "We don’t have any obligations to continue." Their Iceland Airwaves gig would mark their final outing as The Knife. 

Given that the album polarised opinion, it's no surprise that the live shows did too, with the "live" element removed altogether. Instead, a turquoise troupe of dancers gyrate to a CD playback, with hardly an instrument lifted or a vocal sung. The Knife wanted to confront the staleness of live performance and reinject the idea of theatre into music. And even as Karin and Olof went through the moves for their last-ever show, it remains so much fun.

The dancers operate democratically, switching "lead" roles effortlessly, draped in neons and covered in glitter, choreographing the audience as they go. Their interactive techno-ballet represents a genuine defection to an avant garde style that doesn't alienate. Inclusivity is at the heart of this show, and the Dreijers have been extremely vocal about the album's feminist agenda. Halfway through the performance, Karin Dreijer declares: "I want a body I can live in. I want a body with two dicks. I want a body that can't be kicked out of bedrooms, and then I want no bedrooms. I want a body that can fuck and be fucked."

The "deep aerobics instructor" that begins the performance encourages "boys to bounce their balls and ladies bounce your labias, bounce your fucking auras!" and leads a chant chastising self-consciousness. Beyond the obvious novelty of it being their final hurrah, this feels so much more than just a gig – and in playing less, they somehow end up doing so much more.

Material from Shaking the Habitual features most prominently in the set. "Full of Fire" and "A Tooth for an Eye" stands out, as does Karin Dreijer's message to the audience, screaming "without you my life would be boring" before launching into the single of the same name. They close with an amped up version of "Pass This On" from Deep Cuts, reworked to mirror the aesthetic of their new direction.

The Knife's final show takes place at Harpa, Iceland's largest concert hall, that sits on Reykjavík's waterfront. After the show, word goes around that the northern lights had appeared over the venue during the performance, the first time that had happened over the duration of the five-day festival. There is a pleasant, spiritual symmetry to this: both The Knife and the aurora acting as unbridled forces of nature, determined to show the world some wonder.

Read The Knife's last interview with Dazed