  We reduce certain proofs in math.RA/0108067, math.RA/0408155, and
math.QA/0409589 to depth two quasibases from one side only, a minimalistic
approach which leads to a characterization of Galois extensions for finite
projective bialgebroids without the Frobenius extension property. We prove that
a proper algebra extension is a left $T$-Galois extension for some right finite
projective left bialgebroid over some algebra $R$ if and only if it is a left
depth two and left balanced extension. Exchanging left and right in this
statement, we have a characterization of right Galois extensions for left
finite projective right bialgebroids. Looking to examples of depth two, we
establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois
extension. We characterize finite weak Hopf-Galois extensions using an
alternate Galois canonical mapping with several corollaries: that these are
depth two and that surjectivity of the Galois mapping implies its bijectivity.
