Hi guru kido
Some things are not properly formulated, and if I execute the worksheet you posted, I do not see what you show as output. First, in (1) you input things like u11, u22 before having defined them (you define them in (2) -> (6)). But the output of (1) shows u11, u22, ... already defined as in (2) -> (6), so you are doing something else that is not shown here, the order of execution of your worksheet is not from top to bottom. Then, think about: if you input W[i, j, k, l, nonzero], what do you see? You do not show it, but you'd see an equation where the right-hand side is a set of equations. You cannot do arithmetics with such an object, so your Weyl[~i,j,k,l,nonzer] + W[~i,j,k,l,nonzero] has no meaning, it is a nonsensical operation. Your input "halfflatmetrics in 4 D" has no natural sense too. Guru kido, these commands are powerful but you need to read their help pages to understand how to use it. An approach merely looking for a result is of no help; some understanding is.
So answering your question skipping these problems I see in your worksheet: suppose you have two tensors J[a,b,c,d] and K[a,b,c,d] and you want to show that their addition or subtraction (cannot be both, of course) is equal to 0. How do you do that? It is easy: take one minus the other one and send the expression to TensorArray, as in TensorArray(J[a,b,c,d] - K[a,b,c,d], simplifier=simplify). Now you see these are tensors of 4 indices so you cannot see a "matrix" for output. But you can use ArrayElems (take a look at the help pages of these commands before proceeding), so your input will be ArrayElems(TensorArray (J[a,b,c,d] - K[a,b,c,d], simplifier=simplify)), and if J and K are equal then the output will be an empty set; and if the output is not an empty set then they are not equal. I hope this helps you.
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft