  We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl
operator associated with the Weyl group of the exceptional root system $G_2$.
For this symmetry algebra, we give both an abstract definition and an explicit
realisation. We then construct ladder operators, using an intermediate result
we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite
reflection group acting on a three-dimensional space.
