The classification of real Clifford algebras in terms of matrix algebras iswell--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 whenthe square of the volume element $\omega$ is -1, in which case it is uniquelygiven up to sign by right multiplication with $\omega$.
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