The morphism of schemes Spec C->Spec R induced by the inclusion R->C is finite étale. This makes C an étale real algebra. We shall develop a few consequences for real C-C-bimodules. Let A be a four-dimensional real division algebra that is also a bimodule over a subalgebra C isomorphic to the complex numbers. Then, we can show two facts: firstly, its centraliser is one-dimensional; secondly, if A admits an automorphism of order 2, then A becomes a Lie algebra with respect to the Lie bracket [·, ·]. More will follow.