We are interested in the following problem (which may be stated in a much greater generality):

Which algebraic groups ``preserve model-completeness''?

This is supposed to mean that for a given algebraic group G, we want to check whether if K is a model complete field then the (pure) group of rational points G(K) is model complete as well.

It is quite easy to see that the multiplicative group does not preserve model-completeness.

We will discuss the following two positive results.

1. Split semisimple groups preserve model-completeness (joint with Daniel Max Hoffmann, Chieu-Minh Tran, and Jinhe Ye).

2. The Heisenberg group preserves model-completeness (based on the Bachelor Thesis of Maciej Frącek).