Vanishing of Chern-Simons invariant

Question

I am computing Chern-Simons invariants using the metric given by the Bianchi Types, associated naturally to seven of the eight Thurston geometries. However, many of them result to be identically zero. Since I am just testing my code, I am not sure if this is a correct result. I only have that the code does compute correctly the invariant for $\mathbb{S}^3$ with its standard metric, but I do not have other results for 3-mflds to really check if my code works. Would there be any reason to believe this vanishing is true? Especially that $\mathbb{H}^3$, and $\mathbb{H}^2\times\mathbb{R}$ with their std metrics by Bianchi classification turn out to have invariant $\Phi = 0$.