  The decomposition of polynomials of one vector variable into irreducible
modules for the orthogonal group is a crucial result in harmonic analysis which
makes use of the Howe duality theorem and leads to the study of spherical
harmonics. The aim of the present paper is to describe a decomposition of
polynomials in two vector variables and to obtain projection operators on each
of the irreducible components. To do so, a particular transvector algebra will
be used as a new dual partner for the orthogonal group leading to a
generalisation of the classical Howe duality. The results are subsequently used
to obtain explicit projection operators and formulas for integration of
polynomials over the associated Stiefel manifold.
