In Richard Hamilton's first paper on the Ricci Flow, there's a proposition (Lemma 11.6; see page 34 of this document https://projecteuclid.org/download/pdf_1/euclid.jdg/1214436922) who's proof involves taking the tensor $\nabla Rc$ and splitting it into certain components: $$\nabla_iR_{jk}=E_{ijk}+F_{ijk} $$
The essential point is that $F$ is trace-free with respect to any two indices. So $F$ is to $\nabla Rc$ what the Weyl curvature tensor is to the Riemann curvature tensor. (I imagine Hamilton had something like this analogy in mind when thinking this through.)
One can reason in an ad hoc way as follows: Based on the traces of $\nabla Rc$ and the symmetry of the $jk$ indices, one can guess that $$E_{ijk}=a\nabla_iRg_{jk}+b(\nabla_jRg_{ik}+\nabla_kRg_{ij})$$ and solve for the constants $a$ and $b$ using that $F$ should be trace-free.
My question is whether there is a well-established procedure or theorem which tells you how any 3-tensor, or even k-tensor, breaks up into these more fundamental components. For example, I'm certainly aware of how this works for a 2-tensor: $$a_{ij}=\textstyle\frac{tr(a)}{n}g_{ij}+ [\textstyle\frac{1}{2}(a_{ij}+a_{ji})-\textstyle\frac{tr(a)}{n}g_{ij}]+\textstyle\frac{1}{2}(a_{ij}-a_{ji})$$ and that these components are irreducible in the sense of the representation theory of the orthogonal group on tensors. (It's my suspicion that the decomposition $E+F$ above is not irreducible in this sense, as I would naively expect more symmetries at the level of indices.)
I've just chanced across an answer to my question.
The tensor $\nabla Rc$ lies in the kernel of two natural linear maps corresponding to the identities $$\nabla_iR_{jk}=\nabla_iR_{kj}$$ $$\nabla_iR_{ij}=\textstyle\frac{1}{2}\nabla_jR_{ii}$$
(I use the extended Einstein summation convention that repeated lower indices indicate an implicit contraction with the inverse metric.)
In Chapter 16 of Arthur L. Besse's monograph, "Einstein Manifolds" (Generalizations of the Einstein Condition, written by Andrzej Derdzinski) the second section contains a summary of how the linear subspace of $\otimes^3T^*M$ defined by these symmetries decomposes into irreducible components under the action of the orthogonal group. The actual proof that these are the irreducible components uses "standard arguments" in invariant theory and is contained in a paper of Alfred Gray, titled "Einstein-Like Manifolds Which Are Not Einstein"; see section 3 of that paper.