The Cartesian–Klein–Lie approach extends the Erlangen program

The paper considers coordinates as a representation space for a group action, rather than requiring them to form an algebra. In this approach, Cartesian coordinates and Klein–Lie geometry are combined while retaining group-action structure without making algebraic homomorphism essential.

Cartesian–Klein–Lie conjecture. The combination of Cartesian and Klein–Lie approaches based on the assumption that coordinates are often a representation space for a group action is stronger than the original Erlangen program itself and could go beyond previous limits.

The conjecture proposes a broader framework for noncommutative spaces, illustrated by examples in the paper. Its formulation is qualitative, and the supplied source gives no evidence that it has been proved or refuted.