  In this article, we give all the Weitzenb\"ock-type formulas among the
geometric first order differential operators on the spinor fields with spin
$j+1/2$ over Riemannian spin manifolds of constant curvature. Then we find an
explicit factorization formula of the Laplace operator raised to the power
$j+1$ and understand how the spinor fields with spin $j+1/2$ are related to the
spinors with lower spin. As an application, we calculate the spectra of the
operators on the standard sphere and clarify the relation among the spinors
from the viewpoint of representation theory. Next we study the case of
trace-free symmetric tensor fields with an application to Killing tensor
fields. Lastly we discuss the spinor fields coupled with differential forms and
give a kind of Hodge-de Rham decomposition on spaces of constant curvature.
