Better than an automatic smooth continuous motion would be an "idle animation": after a few seconds swing the torus somewhat less than 45 degrees in a random direction and let it relax back into the original position. This suggests "the torus can move" without suggesting "the torus moves [only] in this specific way." You need contradict not just the user's intuition "Torus stationary" but much more importantly "Torus non-interactive."

Likewise, if someone thought the pieces couldn't be moved, you could idly wiggle the king pawn as a hint to pick it up.

Now, how anybody could look at a chess board and think the pieces were immovable, or look at a 3D torus render and think it was unrotatable, is beyond me. But if it's a real problem, I think idle animation is a good fix.