  Recently, various natural algorithmic problems have been shown to be $\exists
\mathbb{R}$-complete. The reduction relied in many cases on the $\exists
\mathbb{R}$-completeness of the problem ETR-INV, which served as a useful
intermediate problem. Often some strengthening and modification of ETR-INV was
required. This lead to a cluttered situation where no paper included all the
previous details. Here, we give a streamlined exposition in a self-contained
manner. We also explain and prove various universality results regarding
ETR-INV. These notes should not be seen as a research paper with new results.
However, we do describe some refinements of earlier results which might be
useful for future research. We plan to extend and update this exposition as
seems fit.
