I think I understand that the inside leg - when on a bend - is used to encourage the horse to engage and flex the inside hind leg. Can you explain a bit more about the outside rein - particularly the phrase 'support with the outside rein'?
This is a good question, as it is one of those terms that gets bandied about with the assumption that everyone knows what it means as it is self-explanatory - or is it?
I remember from my training in Germany being told to think that;
"The inside leg creates the energy and the outside rein regulates it"
In terms of degrees of bend, speed - through half halts, etc. I like the word 'regulate' as it conjures up the image of constant adjustment.
How often do we hear an instructor yelling "more inside leg; more outside rein" and all we end up seeing is a confused rider (not to mention horse) squeezing harder and longer with the leg and hanging on to the outside rein in a death grip?
The concept of Inside Leg to Outside Hand isn't achieved from the outset, but is something we work towards, in the same way we work towards straightening the horse. Our goals are straightness and collection so the horse needs to have even weight on both sides of the bit and his body in alignment, otherwise we run the risk of making his inherent crookedness - that we're working so hard to alleviate - even worse!
However, assuming <g> that your horse is capable of stepping evenly up into his bridle showing good longitudinal 'stretch' from hind hoof to hand
It may help you to think of the inside leg being 'at the girth' and of the horse is being asked to bend around it as a fixed reference point. The 'job', then, of the outside rein is to elastically maintain the horse's shape around the leg, (by supporting the inside aids) but as I said this takes a while to achieve as first the horse must become laterally supple and 'bendable'. If you were to only use the inside leg without the support from the outside rein (and leg!) the horse would move away from the leg, not move around it and he would also fall out through his outside shoulder. I can understand that the word 'support' may have unwanted connotations, but I believe one of my jobs as a teacher is to use the traditional riding terminology and give back its original meaning and not to reinvent the wheel by coming up with my own exclusive, elitist terminology.
To ride it correctly also requires that the horse can flex laterally at the poll. The ability to flex at the poll and flip the nuchal ligament from right to left happens within a very small movement of left to right flexion. When this works the horse fills out (comes to) the outside rein from the engagement of the inside.
There is no sense of making the horse work this way. It comes from a positioning of our own correctly aligned body. Neither is it just the inside leg and outside rein doing all the work! They are but individual instruments in the orchestra of aids that combine to form the symphony of harmonious riding.