  This paper has two objectives. First, we study lattices with skew-Hermitian
forms over division algebras with positive involutions. For division algebras
of Albert types I and II, we show that such a lattice contains an "orthogonal"
basis for a sublattice of effectively bounded index. Second, we apply this
result to obtain new results in the field of unlikely intersections. More
specifically, we prove the Zilber-Pink conjecture for the intersection of
curves with special subvarieties of simple PEL type I and II under a large
Galois orbits conjecture. We also prove this Galois orbits conjecture for
certain cases of type II.
