  In this note we reduce certain proofs in \cite{KS, Karl, AMA} to depth two
quasibases from one side only. This minimalistic approach leads to a
characterization of Galois extensions for finite projective bialgebroids
without the Frobenius extension property: a proper algebra extension is a left
$T$-Galois extension for some right finite projective left bialgebroid $T$ over
some algebra $R$ if and only if it is of left depth two and left balanced.
Exchanging left and right in this statement, we have also a characterization of
right Galois extensions for left finite projective right bialgebroids. As a
corollary, we obtain insights into split monic Galois mappings and endomorphism
ring theorems for depth two extensions.
