a_{4,2}$
   $a_{4,2}$   $a_{4,2}$       $a_{4,3}$        $-a_{4,0}$       $-a_{4,1}$
   $a_{4,3}$   $a_{4,3}$   $\alpha^2 a_{4,2}$   $-a_{4,1}$   $-\alpha^2 a_{4,0}$

From the table it is clear that we have the claimed decomposition.

We use a similar process for the Hopf algebras giving the Hopf-Galois structures of type $C_8$.

Let $ C_{s,t} $ be defined as in . Then we have $$L[C_{s,t}]^G\cong F^2\times F(\beta\iota)^{d_1}\times F(r\iota,\beta\iota)^{d_1d_2} \mbox{ as $ F $-algebras,}$$ where $r,\iota\in F^{\text{alg}} $ such that $r^2=2$, $\iota^2=-1$ and where $d_1=2/[F(\beta\iota):F]$ and $d_2=2/[F(r\iota,\beta\iota):F(\beta\iota)]$.

Let $\eta = \eta_{s,t}$ as defined in , so that $ C_{s,t} = \langle \eta \rangle $, and let $$\begin{aligned}
b_{0}&=\frac{1}{8}\big(1+\eta+\eta^2+\eta^3+\eta^4+\eta^5+\eta^6+\eta^7\big),\\
b_{1}&=\frac{1}{8}\big(1-\eta+\eta^2-\eta^3+\eta^4-\eta^5+\eta^6-\eta^7\big), \\
b_{2}&=\frac{1}{4}\big(1-\eta^2+\eta^4-\eta^6\big),\\
b_{3}&=\frac{1}{4}\big(\eta-\eta^3+\eta^5-\eta^7\big),\\
b_{4}&=\frac{1}{2}\big(1-\eta^4\big),\\
b_{5}&=\frac{1}{2}\big(\eta^3-\eta^7\big), \\
b_{6}&=\frac{1}{2}\big(\eta^2-\eta^6\big),\\
b_{7}&=\frac{1}{2}\big(\eta-\eta^5\big).\end{aligned}$$

It is easily verified that these $ 8 $ elements of $L[C_{s,t}]$ are linearly independent over $ L $ and so form an $ L $-basis of $ L[C_{s,t}] $. Recall from that the nontrivial $G$-orbits of $ C_{s,t} $ are $\{\eta,\eta^{3}\}$, $\{\eta^{2},\eta^{6}\}$ and $\{\eta^{5},\eta^{7}\}$, all with stabiliser $\langle t \rangle$. From this we see that $ b_{0}, b_{1}, b_{2} $ and $ b_{4} $ are fixed by $ G $, that $ \,^{s}b_{3} = -b_{3} $, $ \,^{s}b_{6}=-b_{6} $, and that $ \,^{s}b_{5}=b_{7} $. Therefore the following linear combinations of the above elements are 