  Assume that the compact Riemannian spin manifold $(M^n,g)$ admits a
$G$-structure with characteristic connection $\nabla$ and parallel
characteristic torsion ($\nabla T=0$), and consider the Dirac operator
$D^{1/3}$ corresponding to the torsion $T/3$. This operator plays an eminent
role in the investigation of such manifolds and includes as special cases
Kostant's ``cubic Dirac operator'' and the Dolbeault operator. In this article,
we describe a general method of computation for lower bounds of the eigenvalues
of $D^{1/3}$ by a clever deformation of the spinorial connection. In order to
get explicit bounds, each geometric structure needs to be investigated
separately; we do this in full generality in dimension 4 and for Sasaki
manifolds in dimension 5.
