  The explicit matrix realizations of the reversion anti-automorphism and the
spin group depend on the set of matrices chosen to represent a basis of 1
-vectors for a given Clifford algebra. On the other hand, there are iterative
procedures to obtain bases of 1-vectors for higher dimensional Clifford
algebras, starting from those for lower dimensional ones. For a basis of
1-vectors for Cl (0, 5), obtained by applying such procedures to the Pauli
basis of 1-vectors for Cl(3,0), we find that the matrix form of reversion
involves neither of the two standard representations of the symplectic bilinear
form. However, by making use of the relation between 4 X 4 real matrices and
the tensor product of the quaternions with themselves, the matrix form of
reversion for this basis of 1-vectors is identified. The corresponding version
of the Lie algebra of the spin group, has useful matrix properties which are
explored. Next, the form of reversion for a basis of 1-vectors for Cl(0,6)
obtained iteratively from Cl(0,0) is obtained. This is then applied to the task
of computing exponentials of 5X 5 and 6X 6 real skew-symmetric matrices in
closed form, by reducing this to the simpler task of computing exponentials of
certain 4X 4 matrices. For the latter purpose closed form expressions for the
minimal polynomials of these 4 X 4 matrices are obtained, without having to
compute their eigenstructure. Finally, a novel representation of Sp(4)is
provided which may be of independent interest. Among the byproducts of this
work are natural interpretations for some members of an orthogonal basis for
M(4, R) provided by the isomorphism with the quaternion tensor product, and a
first principles approach to the spin groups in dimensions five and six.
