  We develop a noncommutative integration method for the Dirac equation in
homogeneous spaces. The Dirac equation with an invariant metric is shown to be
equivalent to a system of equations on a Lie group of transformations of a
homogeneous space. This allows us to effectively apply the noncommutative
integration method of linear partial differential equations on Lie groups. This
method differs from the well-known method of separation of variables and to
some extent can often supplement it. The general structure of the method
developed is illustrated with an example of a homogeneous space which does not
admit separation of variables in the Dirac equation. However, the basis of
exact solutions to the Dirac equation is constructed explicitly by the
noncommutative integration method. Also, we construct a complete set of new
exact solutions to the Dirac equation in the three-dimensional de Sitter
space-time $\mathrm{AdS_{3}}$ using the method developed. The solutions
obtained are found in terms of elementary functions, which is characteristic of
the noncommutative integration method.
