  We represent vector rotation operators in terms of bras or kets of half-angle
exponentials in Clifford (geometric) algebra Cl_{3,0}. We show that SO_3 is a
rotation group and we define the dihedral group D_4 as its finite subgroup. We
use the Euler-Rodrigues formulas to compute the multiplication table of D_4 and
derive its group algebra identities. We take the linear combination of rotation
operators in D_4 to represent the four Fermion matrices in Sakurai, which in
turn we use to decompose any 2x2 matrix. We show that bra and ket operators
generate left- and right-acting matrices, respectively. We also show that the
Pauli spin matrices are not vectors but vector rotation operators, except for
\sigma_2 which requires a subsequent multiplication by the imaginary number i
geometrically interpreted as the unit oriented volume.
