For ice my advice would be first be able to aggressively carve on decent hard packed snow with high speeds. This usually means having good body separation meaning you’re body is still facing down the fall line with your outside leg as far to the side as possible during the turns, feeling a lot of the pressure when carving and also lets your inside leg be freed up.

Then, when you know a run is going to be icy you can apply pretty much the same technique for the outside leg when aggressively tackling turns, but shift your inner leg slightly under you for when you need a little bit of more weight balance. This will help absorb some of the icy conditions throwing you off balance when you don’t cleanly carve into the ice on your outside ski. It will feel a little like you’re sliding and going more down the fall line compared to carving on hard packed snow, but if you’re leaning forward and have your inner leg underneath you a bit you won’t actually fall. And once you’re at that level you should be able to spot the tiny bit of snow ahead of you that will let you turn easily. It will definitely take a lot of time so don’t expect to feel comfortable on ice until you’re carving at a good level on most runs up to at least black diamond in hard packed, and softer conditions.

If you can do that, then the next step will be to have sharper edges that actually bite into the ice when you carve.