  The Folium of Descartes in $\mathbb{K}\times\mathbb{K}$ carries group laws,
defined entirely in terms of algebraic operations over the field $\mathbb{K}$.
The problems discussed in this paper include: normalization of Descartes
Folium, group laws and morphisms, exotic structures, exotic structures, second
exotic structure, some topologies on Descartes Folium, differential structure
on Descartes Folium, first isomorphism of algebraic Lie groups over
$\mathbb{K}$, second isomorphism of algebraic Lie groups over $\mathbb{K}$,
derived structures of algebraic Lie groups, a differential/complex analytic
structure on Descartes Folium, Descartes Folium as a topological field, etc.
For predicting these terms, we focus on methods that exploit diagram
manipulation techniques (as alternatives to algebraic method of proofs). All
our results confirm that the Descartes Folium stores natural group structures,
unsuspected till now.
