Method of the moving coordinate frame in the problem of classifying manifolds in algebraic Klein space

It is proven that the (multiplicative) order of the Cartan image and the coordinate ratios of its Chow point provide a complete system of invariants (arithmetic and rational) of algebraic manifold $V$ with respect to algebraic group $G$ of the birational transformations of the space. These invariants define $V$ uniquely to within transformations of $G$.