The Hamiltonian algebra contains isomorphic copies of the real and complex numbers. However, the reals are a central subalgebra of which makes into a real algebra. This makes identifying in canonical: determines a unique embedding . Yet is not a complex algebra. The goal presently is to outline some of the incongruities of and which may be obscured by the notational overlap of the letter .

Proof. Let be a finite dimensional division ring over an algebraically closed field . This means that is a central subalgebra of . Let and consider . Since is central in , is commutative, and so is a field extension of . But as is a finite dimensional space, so is . As any finite dimensional extension of is algebraic, is an algebraic extension. Yet is algebraically closed so . Thus so in fact .

• In particular, this proposition proves is not a complex algebra.
• Alternatively, from the Wedderburn-Artin theorem we know the only semisimple complex algebra of dimension 2 is . This has proper ideals and so it cannot be the division ring .
• It is also evident that the usual, notationally driven, embedding of into is non-central. That is, embeds as , into . This is not central:
  
• Further evidence of the incompatiblity of and comes from considering polynomials. If is considered as a polynomial over then it has exactly two roots as expected. However, if it is considered as a polynomial over we arrive at 6 obvious roots: . But indeed, given any , , then is also a root. Thus there are an infinite number of roots to . Therefore declaring can be greatly misleading. Such a conflict does not arise for polynomials with real roots since is a central subalgebra.