This says a lot. It seems he's comfortable with skiing in that way, and reverts to it. He has fun doing what he's doing--but maybe he could learn some additional ways to have fun on skis.They worked on his hadn position to get him forward which (in turn) brought his legs to more of a gliding wedge position. Now that he's off skiing with mom, his hand position has dropped and he's back to a big wedge instead of a gliding wedge.
I wonder if in his lessons they might have taught him to put his hands on his knees? This is often something younger children are taught, as a way of quieting down all the random movement and of developing adequate forward pressure and balance. Anyway, it promotes success in skiing the way he's now doing it, but it's time to move beyond that. I wouldn't by any means look at the hand position as an issue in isolation. Changing what Lukas does with his hands should just be a part of getting him to start skiing in a much narrower parallel stance, which is what will put him on the path to developing solid skills for a lifetime.
A challenge in teaching children around the age of seven is that their bodies have recently grown and developed to the point that they have greatly advanced in their physical abilities--but their brains may not have quite realized it yet. They may revert to small child habits.
Teaching a child with Lukas' ingrained habits to ski parallel isn't going to be a matter of incrementally adjusting his style. It's going to be a complete break, a new and different way of moving on snow. Therefore I'd suggest looking for ways to play that are different from anything he's done before, and that will develop that new way of moving--for my money, not so much contrived, story-based games, but activities that are fun in themselves. Start again on the gentlest terrain. Perhaps--set up a narrow set of "gates" with cones, fir branches, ski poles, or whatever's available, and the challenge is just to ski in a straight line and go through the gates without touching on either side. Then perhaps practice traverses in which Lukas needs to be able to stay on edge, maintain adequate speed, and not lose elevation, and the challenge is to aim straight at a target on the side of the run and hit it. (It's basically impossible to do this skiing in a wedge.) Then perhaps take off one ski and do "scooter" skiing, learning to balance on one foot.
As for the hands, it's often good to give a child some object to hold on to. When teaching the youngest children, I often used to go around with pockets full of small stuffed animals. I'd ask a child to hold one out in front of him as an extra set of eyes. Lukas looked good holding that pole. Give him a pole--but only one, and not to be used to stab the snow. Maybe give him the advanced version of the horizontal pole holding exercise, where he has to balance the pole on top of open palms while skiing, and not drop it.
Etc., etc.!