so by this point I’m assuming you understand what I mean in saying the quaternion multiplication in four dimensions looks like two rotations perpendicular to and in sync with each other. 
and now it’s finally time to understand why the function we’re looking at is q1 times P times q2 ，this sort of quaternion sandwich around a point， for a starting example.
let’s focus in the J direction, along the y axis think about what it means to multiply by J from the left and then the inverse of J from the right. so when I multiply by J from the left both of those circles rotate right and the IK circle rotates 90 degrees .
but now if I multiply by negative J from the right, the effect on the circle through 1 and J is completely canceled out right .1 moves back to where it started J moves back to where it started.
but the difference is that the circle through K and I continues rotating, just as it had been before. so to see this with slightly different notation. let’s look at that same process but with angle axis .
so in this case our axis is going to be purely in the J direction and let’s say I rotate 90 degrees ,90 degrees in the J direction right .so all circles have rotated 90. then when I rotate whoops, I need to make this the same axis okay so then when I rotate negative 90 degrees from the right side that I case circle just keeps rotating ,
and in fact let’s view that whole thing but bringing in the ijk sphere ,the circle through I and K is part of that sphere so that tells us part of what’s happening to it ,
and then we also know that J and let me pull up negative J are the you know kind of the poles of that sphere on the axis perpendicular to that circle so where J and negative J go along with where K and I go will fully determine where this sphere goes at least under the constraint that it’s all rigid motion up in the four dimensions that we can’t see so as I rotate this 90 degrees that sphere gets to a point where it’s actually projected just as a plane and then as I rotate negative 90 degrees here it comes back to where it was but it’s reoriented okay and we can do this with smaller angles too let’s say instead I rotated just by 45 degrees so about 45 degrees here okay and we see that the IKE a circle rotated 45 degrees and then I do the negative and since it’s from the other side that I case circle keeps on rotating but the sphere is back in place .
and the rule is basically the same for any other unit quaternion it doesn’t have to be J .you know if we did the entire same thing for I, I might change which circles are easier to look at here instead looking at the circle through one and I and the circle perpendicular to that which is J and K .
as I multiply from the left, it sort of bubbles that sphere out and then as I multiply by the inverse from the right the sphere gets back into position but it’s reorient it it’s been rotated. 
so then, that gives basically the intuition for why a certain quarternion sandwich like this gives us rotation in three dimensions .
what it’s actually doing is a much richer much fuller motion up in four dimensions .
we’re just being very clever about how if you construct a certain multiplication from the left and then a certain multiplication from the right the overall effect just on the ijk sphere  equivalent of the equator of the hyper sphere we can’t see, is to reorient it in place, 
and in the applet if you click this little link here what it’ll do is automatically make it, such that the quaternion on the right is the inverse of the one on the left. 
so as you tweak the angle and the axis you don’t have to worry about hand constructing that right one to be the inverse all right .
so that’s the end of the lesson that I had planned. but don’t let this be the end of your actual interaction here.    
there’s a lot that you can do if you add various different elements on here and I want you to ask yourself questions and see if you can answer those questions just by playing around and maybe popping down to some of the lower dimensions to help build intuition.
for example if you want to understand the angle axis representation here. you know usually when you see a cosine and a sine, that means you’re along some kind of circle. well what circle are you along, see what that looks like in three dimensions. if we represent a point of that sphere ,this little pink dot with a certain angle and axis .
and tell me what circle are you on and how does that look under the stereographic projection. 
likewise ask yourself different questions play around with different elements and I think very seriously you could productively spend hours here maybe even pulling out a pencil and paper to kind of work out some of these products by hand and seeing how it all fits together