The problem with sitting well starts the moment you consider going to sit, well before your derriere hits the chair. To sit well you need to be able to travel to the chair well. The quality of your destination will be marked by the quality of your journey.
As I mention in the video, I'm not actually interested in you doing something correctly, but freely. Ideally I wished I'd expanded on that a bit in the video (the pitfalls of improvised content), so I'll take the opportunity to do so here. To do something freely means to do something without reaction. The most common place we observe that reaction is in the startle reflex, the pulling of the shoulders up, and the pulling of the neck muscles so the head tilts backwards. There may not be actual movement visible in the head/neck relationship, but there will still be an undue tightening of the muscles. You may also observe that your breath becomes held.
What's important here isn't the physicality though, that's just an outer manifestation that there's been a change in your thinking, and that's the reaction. Essentially, going to sit should be as inconsequential as scratching your nose; Wholly unremarkable. It's about moving yourself through space without upsetting the quality of your nervous system and thinking, and in doing so you'll find the movement itself easier. It's like carrying a full glass of water without disturbing the contents so it doesn't spill, where your very being is the water.
I’d go as far to say as the problem with sitting down starts with the phrase “sitting down”. It’s the “down” part of the phrase that’s the issue, it encourages you to think down in a way that encourages collapse. Better to maintain the upwards direction of the head and spine and fold your legs up underneath your torso as the floor pushes up into your feet. There’s a longer conversation to be had there about the true nature of gravity (long story short, it’s not pulling you down!), but this isn’t the place right now to be getting into Einstein’s General Theory of Relativity.
The action of sitting and standing is so rich in exploration of human behaviour that we use it as a contrivance and metaphor for all movement in teaching the Alexander Technique. It's a "vanilla" flavoured movement we use as a teaching tool and and as a starting point. Individual teachers may have a specialist niche in other movements, such as playing an instrument (I play guitar and often work with fellow guitarists ) or sports activities, but the same behaviours and attitudes can be observed in "vanilla" movements as can be seen in more specific ones.
Given our lifestyles though, it's a very useful functional movement to be able to do freely. It's also good training for squatting freely too. If you can't do the first half of going into a squat freely, you can guarantee you you wont be able to do the second half freely either. It'll be repleat with effort and excess tension, robbing you of the benefits a good squat provides for releasing the lower back muscles and hips.
Now, if you'll forgive my sartorial inelegance, I forgot I was still wearing my tracksuit bottoms when I decided to make this video somewhat impromptu, you'll find all need to know with regards to sitting down with poise and control within it.