  Let $S$ be the left bialgebroid $\End {}_BA_B$ over the centralizer $R$ of a
right D2 algebra extension $A \| B$, which is to say that its tensor-square is
isomorphic as $A$-$B$-bimodules to a direct summand of a finite direct sum of
$A$ with itself. We prove that its left endomorphism algebra is a left
$S$-Galois extension of $A^{\rm op}$. As a corollary, endomorphism ring
theorems for D2 and Galois extensions are derived from the D2 characterization
of Galois extension (cf. math.QA/0502188 and math.QA/0409589). We note the
converse that a Frobenius extension satisfying a generator condition is D2 if
its endomorphism algebra extension is D2.
