  We study the decomposition into irreducibles of the kernel of noncubic Dirac
operators attached to finite-dimensional modules. We compare this decomposition
with features of Kostant's cubic Dirac operator. In particular, we show that
the kernel of noncubic Dirac operators need not contain full isotypic
components. The cases of classical and exceptional complex Lie algebras are
studied in detail. As a by-product, we deduce some information on the kernel of
noncubic geometric Dirac operators acting on sections over compact manifolds
studied by Slebarski.
