  We prove quantitative versions of Borel and Harish-Chandra's theorems on
reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on
the lengths of reduced integral vectors in any rational representation of a
reductive group. Secondly, we obtain polynomial bounds in the construction of
fundamental sets for arithmetic subgroups of reductive groups, as the latter
vary in a real conjugacy class of subgroups of a fixed reductive group.
  Our results allow us to apply the Pila--Zannier strategy to the Zilber--Pink
conjecture for the moduli space of principally polarised abelian surfaces.
Building on our previous paper, we prove this conjecture under a Galois orbits
hypothesis. Finally, we establish the Galois orbits hypothesis for points
corresponding to abelian surfaces with quaternionic multiplication, under
certain geometric conditions.
