Kevin Scott from Living Voice UK :
The room is always a significant part of the sound. Only part of what you hear comes directly from the speakers, and the rest is the reverberant field of the room. The sound characteristic of the reverberant field is dictated first by the 'off-axis' behaviour of the speaker, and then by the dimensions and materials that the room is made of and its' contents. There is a complicated science to this - almost unfathomably complicated so an academic approach is best left to professional acousticians - and in my experience they usually revert to the best approach which is intuition with trial and error. Experience obviously helps.
A common mistake is to attack the situation with purely absorptive, deadening room treatment. These invariably operate over a wide bandwidth so that whilst they subdue your target problematic frequency range, they also subdue everything else as well and kill the life and space and freedom in the sound-world.
Sound can be absorbed, reflected, or diffused…the latter two usually bring the results you seek.
Bass traps in the corners can be useful 'IF' there is a bass / standing wave problem. I have heard them make some rooms worse, tread cautiously and make one step at a time.
Put them in and then take them out and see how this feels. Back corners are as effective as front corners, you do not necessarily need to treat all of them. Dealing with mid and HF emphasis or confusion is best done using scattering and diffusion. This is why when we do an exhibition in an unknown room, we like to get there a day or so early to let the system settle down and then we can pay attention to how the room behaves with scattering and diffusion.
Your room has the first reflected wave off the floor damped by a rug. This is usually my first listening experiment and only rarely have I thought it detracted. Usually everything improves.
The glass between and behind the speakers is a nice thing to have. In rooms with heavy drapes, I usually find much better spatial coherence with the drapes left open. It is probably good to experiment with some scattering and absorption at the back of the room as well as in the back corners. Not bass traps but reflective and 'scattery' surfaces angled across the corners such as book cases, LP and CD storage. Acoustic phase arrays that are relatively deeply pocketed…deeper than the phase arrays that you see at the sides of rooms can be very useful and look nice. Inexpensive as well.