  Spherical monogenics can be regarded as a basic tool for the study of
harmonic analysis of the Dirac operator in Euclidean space R^m. They play a
similar role as spherical harmonics do in case of harmonic analysis of the
Laplace operator on R^m. Fix the direct sum R^m = R^p x R^q. In this paper we
will study the decomposition of the space M_n(R^m;C_m) of spherical monogenics
of order n under the action of Spin(p) x Spin(q). As a result we obtain a
Spin(p) x Spin(q)-invariant orthonormal basis for M_n(R^m;C_m). In particular,
using the construction with p = 2 inductively, this yields a new orthonormal
basis for the space M_n(R^m;C_m).
