  It is of interest to characterize algebraically the dynamical types of
isometries of the complex and quaternionic hyperbolic planes. In the complex
case, such a characterization is known from the work of Giraud-Goldman. In this
paper, we offer an algebraic characterization of the isometries of the
two-dimensional quaternionic hyperbolic space. Our result restricts to the
complex case and provides another characterization of the isometries of the
complex hyperbolic plane which is different from the characterization due to
Giraud-Goldman.
  Two elements in a group G are said to be in the same z-class if their
centralizers are conjugate in G. The $z$-classes provide a finite partition of
the isometry group. In this paper we describe the centralizers of the
isometries, and determine the z-classes.
