  The Dunkl deformation of the Dirac operator is part of a realisation of an
orthosymplectic Lie superalgebra inside the tensor product of a rational
Cherednik algebra and a Clifford algebra. The Dunkl total angular momentum
algebra (TAMA) occurs as the supercentraliser, or dual partner, of this Lie
superalgebra. In this paper, we consider the case when the reflection group
associated with the Dunkl operators is a product of two dihedral groups acting
on a four-dimensional Euclidean space. We introduce a subalgebra of the total
angular momentum algebra that admits a triangular decomposition and, in analogy
to the celebrated theory of semisimple Lie algebras, we use this triangular
subalgebra to give precise necessary conditions that a finite-dimensional
irreducible representation must obey, in terms of weights. Furthermore, we
construct a basis for representations of the TAMA with explicit actions.
Examples of these modules occur in the kernel of the Dunkl--Dirac operator in
the context of deformations of Howe dual pairs.
