I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the papers of Bismut-Cheeger and Dai where a formula is given in terms of Bismut-Cheeger eta forms, but I'm having a hard time actually computing them.

Are there places where they are computed explicitly for some $S^{2n}$-bundles? This MO answer gives various explicit computations of eta invariants but I didn't find $S^{2n}$ bundles there.

Edit:

To add a bit more context, I’m interested in the following restricted situation.

Let $Spin(2n)$ act on $S^{2n}$ fixing the north and the south poles. Pick a metric compatible with this action. Now pick an $Spin(2n)$ bundle $P$ with a connection over an odd dimensional base $B$. From this data we have a fibration $S^{2n}\to X\to B$. Then I should be able to compute the eta form of the Dirac operator on $X$.

What I’m confused is that, as the eta form is local on $B$, it should be a local expression of the metric on $B$ and the $Spin(2n)$ connection on $P$. But the eta form is an odd-degree form, and I can’t think of any way to write a local expression in terms of the metric and the $Spin(2n)$ connection! Does it mean that the eta form in this class of examples is simply zero?

I would also like to consider the case when we have an $Spin(2n)$ equivariant vector bundle $E$ over $X$ and compute the eta form for the Dirac operator tensored with $E$. I’m still at a loss what nonzero term I can write in this situation.