Diffeomorphism invariant scalars of a Riemannian manifold

Question

Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives:

1. Ricci scalar $R$
2. $R_{ab}R^{ab}$
3. $R_{abcd}R^{abcd}$

Are there more independent scalars that can be constructed or are all diffeomorphism invariant scalars functions of these?

What about $R_{ab}R_{cd}R^{acbd}$?

I am looking for a list of independent scalars such that any other scalar can be expressed as a function of scalars in the list.

Any relevent references are most welcome.