Let $B$ be a ring with 1 and $C$ the center of $B$. It is shown that
if $B$ is a Galois algebra over $R$ with a finite Galois group
$G$, $J_g=\{b\in B\,|\,bx=g(x)b$ for all $x\in B\}$ for each
$g\in G$, and $e_g$ an idempotent in $C$ such that $BJ_g=Be_g$,
then the algebra $B(g)$ generated by $\{J_h\,|\,h\in G$ and
$e_h=e_g\}$ for an $g\in G$ is a separable algebra over $Re_g$ and
a central weakly Galois algebra with Galois group $K(g)$
generated by $\{h\in G\,|\,e_h=e_g\}$. Moreover, $\{B(g)\,|\,g\in
G\}$ and $\{K(g)\,|\,g\in G\}$ are in a one-to-one
correspondence, and three characterizations of a Galois extension
are also given.