Complex structure of a real Clifford algebra

The classification of real Clifford algebras in terms of matrix algebras is well--known. Here we consider the real Clifford algebra ${\mathcal Cl}(r,s)$ not as a matrix algebra, but as a Clifford module over itself. We show that ${\mathcal Cl}(r,s)$ possesses a basis independent complex structure only when the square of the volume element $\omega$ is -1, in which case it is uniquely given up to sign by right multiplication with $\omega$.