Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.

Let $W \subset \Lambda^2V^* \odot \Lambda^2V^*$ be the subset satisfying the algebraic Bianchi identity. More precisely, let $R(v_1,v_2,v_3,v_4)$ denote an element of $\Lambda^2V^* \odot \Lambda^2V^*$. Thus $R$ is skew-symmetric in $v_1$ and $v_2$ and it is also skew-symmetric in $v_3$ and $v_4$. Moreover

$$ R(v_3,v_4,v_1,v_2) = R(v_1,v_2,v_3,v_4). $$

Then $R \in W$ if and only if, in addition to the conditions above, $R$ also satisfies the following identity (known as the algebraic Bianchi identity):

$$ R(v_1,v_2,v_3,v_4) + R(v_2,v_3,v_1,v_4) + R(v_3,v_1,v_2,v_4) = 0. $$

Now my question can be formulated. What is an explicit description of the ring $\mathbb{C}^G[W]$ of $G$-invariant polynomials on $W$ (with $W$ being the space of algebraic curvature tensors, if I may call it so) and $G$ acting on $W$ by restricting its natural action on $\Lambda^2V^* \odot \Lambda^2V^*$.

Also, if one fixes a degree $d > 0$, what is an explicit description of the space of $G$-invariant homogeneous polynomials in $W$ of degree $d$?

I was thinking at first about the Chern-Weil homomorphism, but I think this only gives a proper subspace of $G$-invariant polynomials on $W$, and not all of them (I am not a 100% sure). This has probably been studied in the literature. I don't have access to MathSciNet anymore though (due to some budget cuts by my University).

Edit 1: I notice some overlap with the post Invariant polynomials in curvature tensor vs. characteristic classes, but the posts are sufficiently different.