I have more specific advice. Forgive my non-technical explanation:
1. The hip to torso angle is key. In other words, upright torso but angulated femur. Our race coaches tell the kids to imagine trying to squish a doughnut on the outer side of their outer hips. This helps transmit downward pressure on the inner edge of the outer ski, so important in holding your edge on an icy race course. I don't teach so I'm not sure if this is what is referred to as angulation. The common mistake of leaning into the hill (which looks great to untrained eyes) with legs and torso lined up straight, even with decent edge angles, is deadly on an icy course. We race dads call it "leaning" and it always results in a fall or a lot of sliding around on the ice.
2. A -framing is another enemy of carving on ice. Racers are sometimes told to "drop their inner knee" to correct this. It makes for a more powerful turn and definitely enhances your hold on the ice. Gotta separate your knees a bit.
3. A corollary of 1 and 2, you've gotta put your weight on your outer ski.
4. Sharp skis of course. Most shops can't properly tune for New England ice, and tuning needs to be done every day or two when it's really firm. So you really have to learn to do it yourself if you ski frequently in the Northeast, and it takes a lot effort. After 6 years with 3 kids racing, I'm still learning. One common piece of bad advice is that it's good enough to just touch up the edges with a fine stone after a day on the ice. Not true! If the edges are dull, you need to take a little metal off. This requires a file, or at least a lot of pressure with a 100 stone, then the usual stone progression.
5. Also helps to have race-oriented or race-inspired skis with less than 85 mm underfoot, but no need for real race skis.
6. When it's really blue ice and nothing works, be light on your skis, take a little pressure off, glide a bit.
7. Be wary of tuning and technical advice from anyone knows nothing about racing. Especially out West, where the only ice anyone seems to ski is on the race hill. Some get it and some don't. Plenty of nice-looking skiers at the great hills out west are ripping up the soft stuff but they avoid hard snow. Often it's because they are leaning (see #1) and can't hold an edge on ice. A great skier should enjoy all conditions (spoken like a true Easterner!!)
Apologies if this has been said, but I don't think I read a post that put it in these layman's terms. It's just my interpretation of the great race coaching my kids have gotten over the years, and some instructors may take issue. Focusing on these points has definitely enhanced my skiing.