I have been doing some work on the diamond material. It is already pretty amazing but I am trying to make it more like your example. I still need to study more on nodes but I will show you what I have found so far.
I made a few changes to the Blend4Web stock diamond material example. It has two meshes, and inside and an outside. I deleted the inside mash (back up your original) then duplicated the outside mesh and left it in the same place. Then, in edit mode I flipped all the normals and made a simple reflective material. I activated Reflective in the Object tab.
Then, I made a change to the original diamond material.
The part shown here is in the gem_outside_material-core. Here is a video that shows you how to access the inner core materials using the TAB key:https://youtu.be/DswQhRFo6OQ
But so far, the diamond in your example is still much nicer. I am going to spend some more time learning the material nodes and see if I can do better.