Let $\F$ denote either of $\R$, $\C$ or the quaternions $\H$. Let $H^2_{\F}$
denote the two dimensional hyperbolic space over $\F$. The algebraic
characterization of the isometries of $H^2_{\R}$ and $H^3_{\R}$ in terms of
their trace and determinant are foundational in the real hyperbolic geometry.
The counterpart of this characterization for isometries of $H^2_{\C}$ was given
by Giraud and Goldman. In this paper we offer algebraic characterization for
the isometries of $H^2_{\H}$. The methods we follow carry over to the complex
hyperbolic space, and yields an alternative characterization of the isometries
of $H^2_{\C}$ which is different from those of 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 of isometries of the real
hyperbolic space have been described by Gongopadhyay-Kulkarni. In this paper we
describe the $z$-classes of isometries of the two-dimensional complex and
quaternionic hyperbolic space. In fact, we explicitly compute their number.