where $\eta$ is the suspension of the hopf fibration.
When I say "in coordinates" I mean that $2\eta$ comes from choosing an explicit representation of $\eta :S^3\to S^2$, suspending it, composing with an explicit map of order 2 and then describing it's null-homotopy in coordinates as well.
You can work this out from the Pontryagin-Thom description. $\eta$ is given by a figure-8 framed unlink in 3-space, suspending it means embedding $\mathbb{R}^3$ into $\mathbb{R}^4$ in the obvious way. $-\eta$ is given by a figure-8 unlink with the undercrossing. A bordism between these $1$-manifolds is given by switching an over-crossing to an under-crossing by moving a little into the 4th dimension so that you can move it past and then going back. I've just described a framed bordism in $\mathbb{R}^4$ and applying Pontryagin-Thom will give you a homotopy between $\eta$ and $-\eta$.
Turning this into a null-homotopy of $2 \eta$ instead of a homotopy of $\eta$ with $-\eta$ is slightly annoying, but just a matter of whiskering with $\eta$ on both sides.
Translating all this into explicit coordinates is certainly tedious, but should be straightforward. It's unlikely to me that seeing the calculation in coordinates will be more illuminating than just thinking about the bordism itself.
