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

Classical stability of black holes under massless Dirac perturbations