This first clip shows us a slightly older Rita Hayworth steaming up the screen in the Caribe-noir film The Fire Down Below, costarring Bob Mitchum (the undisputed king of all Calypso) and Jack Lemmon (inspiration for Ol' Gil). While the direction's pretty uneven, that same unevenness provides license for this kind of over-the-top dance number, opportunistically echoing the feverish performers limboing in the opening of the film. I think it's supposed to be "look how evil she is - dancin' all sexy like that - she's tryin' to seduce Mitchum!" but it's more like "huh?!?!?" Regardless- she dances her ass off, it grips one, and is absolutely worth setting your mind free with.
This one is completely different but just as why-you-watch-dance gripping. From Band of Outsiders, it's a means for Godard to set the mood for danger and tension to come by doing the opposite of danger & tension, underscoring the fun of the scene with commentary that abruptly stops only the music, not the wild track, and states in different ways just what tabulae rasa these fools playing at gangster are by describing feelings that have nothing to do with robbery. Quentin Tarantino named the production company A Band Apart after this film and it's easy to see why - Godard plays with the tools that are used to build a narrative in unconventional ways- breaking rules, even by today's standards, in order to step outside the traditional narrative to reinforce it. Plus cool music, and a great dance routine in a bleak mise-en-scene.
Get it here, BTW.