ArrayElems@TensorArray returns different than an empty set when the tensorial expression you passed is not equal to zero. If you were expecting zero and are not receiving that, you'd need to revise your formulation. For example, if the Weyl tensor is (equal to its) anti-self dual, then the Bach and Eastwood–Dighton tensors vanish (see for instance Self–Dual Conformal Gravity). You can check whether these tensors vanish or not, and if they don't you have no reason to expect C + W = 0.
Your "computation of more derivatives" (equations (12) to (18)) has nothing to do with the result being or not being zero. I see, however, that your equation (11) says that some combination of third order derivatives of u(X) is equal to zero. If that is your case and you want to simplify your nonzero result taking into account identities like (11), then the way to do that is to use simplify - see the help page ?simplify,siderels
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft
PS: when you want to show that something is not zero you do not need to present pages and pages of output; in case like that one just say "the following does not return zero". Otherwise, it is difficult to read your posts. Likewise, you were aware of the "misordering" but nevertheless posted that way. That does not help. You need to be more precise with your posts or otherwise it becomes too time consuming to give you some feedback.