  All iterated skew polynomial extensions arising from quantized universal
enveloping algebras of Kac-Moody algebras are special examples of a very large,
axiomatically defined class of algebras, called CGL extensions. For the
purposes of constructing initial clusters for quantum cluster algebra
structures on an algebra R, and classification of the automorphisms of R, one
needs embeddings of R into quantum tori T which have the property that R
contains the corresponding quantum affine space algebra A. We explicitly
construct such an embedding A \subseteq R \subset T for each CGL extension R
using the methods of noncommutative noetherian unique factorization domains and
running a Gelfand-Tsetlin type procedure with normal, instead of central
elements. Along the way we classify the homogeneous prime elements of all CGL
extensions and we prove that each CGL extension R has an associated maximal
torus which covers the automorphisms of R corresponding to all normal elements.
For symmetric CGL extensions, we describe the relationship between our quantum
affine space algebra A and Cauchon's quantum affine space algebra generated by
elements obtained via deleting derivations.
