A large number of Riemann tensor invariants can be written as traces of products of two 3×3 matrices, representing the Weyl tensor and the Weyl-like square of the Ricci tensor. It is pointed out that finding a complete set, ℐ, for all of these invariants is a simple consequence of the more general problem of finding a complete set of symmetric matrices, ℳ, in terms of which all symmetric matrix polynomials in these two matrices can be expressed. Such a set is constructed and a formal proof of its completeness is given. Several matrix identities and a scalar syzygy, obtained recently by Sneddon, are rederived and their interrelationships clarified. They are shown to be, ultimately, consequences of the Cayley–Hamilton theorem. A “minimal set” of invariants, that must be contained in the complete set of invariants of the general problem, is identified and it is concluded that no set proposed so far is complete.