  We investigate Diophantine definability and decidability over some subrings
of algebraic numbers contained in quadratic extensions of totally real
algebraic extensions of $\mathbb Q$. Among other results we prove the
following. The big subring definability and undecidability results previously
shown by the author to hold over totally complex extensions of degree 2 of
totally real number fields, are shown to hold for {\it all} extensions of
degree 2 of totally real number fields. The definability and undecidability
results for integral closures of ``small'' and ``big'' subrings of number
fields in the infinite algebraic extensions of $\mathbb Q$, previously shown by
the author to hold for totally real fields, are extended to a large class of
extensions of degree 2 of totally real fields. This class includes infinite
cyclotomics and abelian extensions with finitely many ramified rational primes.
