all fixed by $G$, and in fact form a basis of $L[C_{s,t}]$ over $L$:

$$\begin{aligned}
a_{0}&=b_{0},\\
a_{1}&=b_{1},\\
a_{2,0}&=b_{2},\\
a_{2,1}&=\beta b_{3}=\beta b_{2}\eta, \\
a_{3,0}&=b_{4}=\mathfrak{e}, \\
a_{3,1}&=\beta b_{6}=\beta\mathfrak{e}\eta^2,\\
a_{3,2}&=(b_{5}+b_{7}) = \mathfrak{e}(\eta^{3}+\eta), \\
a_{3,3}&=\beta(b_{5}-b_{7}) = \beta \mathfrak{e}(\eta^{3}-\eta). \end{aligned}$$

We have $ {a}_i{a}_j = \delta_{i,j} {a}_i $ for $ i,j=0,1 $, ${a}_i{a}_{2,k}=0$ for $i=0,1$ and $k=0,1$, ${a}_i{a}_{3,k}=0$ for $i=0,1$ and $k=0,1,2,3$, and $ a_{2,k}a_{3,l}=0 $ for $ k=0,1 $ and $ l=0,1,2,3 $. Finally, we consider the multiplication tables of the $ a_{2,k} $ and the $a_{3,k}$.

               $a_{2,0}$       $a_{2,1}$
  ----------- ----------- --------------------
   $a_{2,0}$   $a_{2,0}$       $a_{2,1}$
   $a_{2,1}$   $a_{2,1}$   $-\beta^2 a_{2,0}$

               $a_{3,0}$        $a_{3,1}$          $a_{3,2}$          $a_{3,3}$
  ----------- ----------- ---------------------- -------------- ----------------------
   $a_{3,0}$   $a_{3,0}$        $a_{3,1}$          $a_{3,2}$          $a_{3,3}$
   $a_{3,1}$   $a_{3,1}$   $-\beta^{2} a_{3,0}$    $a_{3,3}$     $-\beta^{2} a_{3,2}$
   $a_{3,2}$   $a_{3,2}$        $a_{3,3}$         $-2 a_{3,0}$       $-2a_{3,1}$
   $a_{3,3}$   $a_{3,3}$   $-\beta^{2} a_{3,2}$   $-2a_{3,1}$     $2\beta^2 a_{3,0}$

From these tables it is clear that we have the claimed decomposition.

Comparing these results with those obtained in section 3, we see that two Hopf algebras giving Hopf-Galois structures of the same abelian type on $ L/F $ are isomorphic as Hopf algebras if and only if they are isomorphic as $ F $-algebras. On the other hand, although Hopf algebras giving Hopf-Galois structures of different types are not isomorphic 