  It is shown that the free Dirac equation in spherically symmetric static
backgrounds of any dimensions can be put in a simple form using a special
version of Cartesian gauge in Cartesian coordinates. This is manifestly
covariant under the transformations of the isometry group so that the
generalized spherical coordinates can be separated in terms of angular spinors
like in the flat case, obtaining a pair of radial equations. In this approach
the equation of the free field Dirac in $AdS_{d+1}$ backgrounds is analytically
solved obtaining the formula of the energy levels and the corresponding
normalized eigenspinors.
