# Importance of Tracelessness of Tensor?

What makes the trace-free tensor (or part of it) so important? As in trace-free Ricci tensor or Weyl tensor.

Symmetry. A general rank-2 tensor $$T\in V\otimes V$$ can be decomposed as
$$T = \tau I + S + A,$$
where $$n\tau = \operatorname{Tr}(T)$$ ($$n$$ being the dimension of $$V$$), $$I$$ is the identity, $$S$$ is trace-less symmetric and $$A$$ is skew-symmetric. These "components" obey different transformation laws under a given symmetry. For example, the scalar part $$\tau I$$ is obviously invariant.