  We introduce the symplectic group $\mathrm{Sp}_2(A,\sigma)$ over a
noncommutative algebra $A$ with an anti-involution $\sigma$. We realize several
classical Lie groups as $\mathrm{Sp}_2$ over various noncommutative algebras,
which provides new insights into their structure theory.
  We construct several geometric spaces, on which the groups
$\mathrm{Sp}_2(A,\sigma)$ act. We introduce the space of isotropic $A$-lines,
which generalizes the projective line. We describe the action of
$\mathrm{Sp}_2(A,\sigma)$ on isotropic $A$-lines, generalize the
Kashiwara-Maslov index of triples and the cross ratio of quadruples of
isotropic $A$-lines as invariants of this action. When the algebra $A$ is
Hermitian or the complexification of a Hermitian algebra, we introduce the
symmetric space $X_{\mathrm{Sp}_2(A,\sigma)}$, and construct different models
of this space.
  Applying this to classical Hermitian Lie groups of tube type (realized as
$\mathrm{Sp}_2(A,\sigma)$) and their complexifications, we obtain different
models of the symmetric space as noncommutative generalizations of models of
the hyperbolic plane and of the three-dimensional hyperbolic space.
  We also provide a partial classification of Hermitian algebras in Appendix A.
