  In a D-dimensional maximally symmetric spacetime we simplify the massless
Dirac equation to two decoupled wavelike equations with effective potentials.
Furthermore in D-dimensional Schwarzschild and Schwarzschild de Sitter black
holes we note that for the massless Dirac field moving in the region exterior
to the event horizon at least one of the effective potentials is not positive
definite. Therefore the classical stability of these black holes against this
field is not guaranteed. Here with the help of the S-deformation method, we
state their classical stability against the massless Dirac field, extend these
results to maximally symmetric black holes, and comment on the applicability of
our results to establish the stability with respect to other classical fields.
