  The decomposition of the spinor bundle of the spin Grassmann manifolds
$G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of
$\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction
is developed and the general statement is proven for $G_{2k+1,3}$, $G_{2k,4}$,
and $G_{2k+1,5}$ for all $k$. The decomposition is used to discuss properties
of the spectrum and the eigenspaces of the Dirac operator.
