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The basic concept behind shooting reflective objects to get the illusion of 3D on then is to control and light the space around them which is seen in the mirror reflections on the object, which are analogous perceptually to the highlights a direct source would create on the object. That takes care of half of creating the illusion of 3D in a 2D rendering: the highest / nearset points to the apparent source of light, which appears more natural when the reflections of the background are on the upper quadrant of the curves where mid-day sun would put them.
The other half the "make it look 3D" optical/perceptual illusion is creating the impression that other parts of the object are lower than those where the highlights are located. That's done by blocking the reflections the white surround is creating on the object by placing "flags" such as black cards or draping to "kill" the reflections on the same areas where you'd expect to see shadows; opposite the position of the highiights. If the highlights are at 2 o' clock on top you'd expect to see the shadows at 8 o' clock.
But your situation isn't typical because you don't want the reflections which define the outer clear glass ball to get in the way of seeing the interior detail. Because of that I'd suggest you start with lights bounced off walls and ceiling of the room to create a diffuse source with a natural downward vector then surround the object with black draping, including over the camera so it's reflection isn't seen. The selectively add white cards to create the high point highlights on the object. Starting with all black and then adding the highlights will make it easier to control where they wind up on the outer sphere.
The Catcb-22 is that the ideal lighting strategy for minimizing the reflection on the outside may not be the best one for rendering the 3D shape of the interior. There are tricks to get around that problem, such as cutting a "keyhole" in the background or above / below which isn't visible from the camera position to selectively light the interior with a narrow beam of direct light from a source behind the backdrop.
But all things considered the best strategy will probably be to do a composite. Shoot from a tripod and take one exposure with the lighting optimized to eliminate any reflections on the outer ball seen over the interior of the ornament, then another opitmized to make the exterior look like a 3D ball via reflections, and possibly a third with the "key hole" strategy to optimally render the interior. Then layer the three in Photoshop with the exterior 3D reflection shot on the bottom and interior on top using a mask blend in the image of the interior over any reflections on the exterior shot that where blocking it on the base layer.
Blending the layers by eye with masks and the opacity sliders you'll be able to recreate in the photo how your brain when looking at them in person sees reflections on the outside ball but mentally "tunes them out". Thus, you don't want to eliminate all reflections over the interior (which will look fake), just cut them back to about 20-30% of the way the camera recorded them until they look "right".