  The well-known classification of the Clifford algebras $Cl(r,s)$ leads to
canonical forms of complex and real representations which are essentially
unique by virtue of the Wedderburn theorem. For $s\ge 1$ representations of
$Cl(r,s)$ on $R^{2N}$ are obtained from representations on $R^N$ by adding two
new generators while in passing from a representation of $Cl(p,0)$ on $R^N$ to
a representation of $Cl(r,0)$ on $R^{2N}$ the number of generators that can be
added is either 1, 2 or 4, according as the Clifford algebra represented on
$R^N$ is of real, complex or quaternionic type. We have expressed canonical
forms of these representations in terms of the complex and quaternionic
structures in the half dimension and we obtained algorithms for transforming
any given representation of $Cl(r,s)$ to a canonical form. Our algorithm for
the transformation of the representations of $Cl(8d+c,0)$, $c\le 7$ to
canonical forms is based on finding an abelian subalgebra of $Cl(8d+c,0)$ and
its invariant subspace. Computer programs for determining explicitly the change
of basis matrix for the transformation to canonical forms are given for lower
dimensions. The construction of the change of basis matrices uniquely up to the
commutant provides a constructive proof of the uniqueness properties of the
representations and may have applications in computer graphics and robotics.
