Wikipedia has explicit formulas for first and second degree polynomial integrals in the unitary group using Weingarten functions (https://en.wikipedia.org/wiki/Weingarten_function).
Is there any closed form formula for similar integrals but in the Orthogonal group $O_d$? I am refering to integrals of the form:
$\int_{O_d} o_{ij} o_{kl} o_{mn} o_{pq} d\mu(O_d)$
where $\mu(O_d)$ is the Haar measure of the Orthogonal group.
I have found Theorem 2.1 and Proposition 2.2 in https://arxiv.org/pdf/0910.1258.pdf but I'm not sure how to go from this to a closed form formula, similar to the Wikipedia article. Thanks!