05:52
are you actually talking about the matrix $\mathbb T$ in section 7.1.2 of your link?
In that case you're looking for the "volumetric-deviatoric decomposition" of the stress (not stress-energy) tensor
The volumetric component ($-p\mathbb I$) is the trace part, because it contains the normal stress in each direction on an infinitesimal fluid packet at rest (it should be something like $\frac13\delta_{ij}\sum_k \sigma_{kk}$), and the trace is coordinate-invariant
Since the deviatoric component ($\mathbb T$) is the remaining part, it's clearly traceless
