  It is shown that, for spherically symmetric static backgrounds, a simple
reduced Dirac equation can be obtained by using the Cartesian tetrad gauge in
Cartesian holonomic coordinates. This equation is manifestly covariant under
rotations so that the spherical coordinates can be separated in terms of
angular spinors like in special relativity, obtaining a pair of radial
equations and a specific form of the radial scalar product. As an example, we
analytically solve the anti-de Sitter oscillator giving the formula of the
energy levels and the form of the corresponding eigenspinors.
