as Hopf, in certain situations it is possible that they are isomorphic as $ F $-algebras. For example: if $ \beta = \iota $ then $ L[E_{s,t}]^{G} \cong L[A_{s,t}]^{G} $ as $ F $-algebras.

The remaining structures are of nonabelian type, and so we cannot employ the methods of [@WATERHOUSE §6.3] or [@BLE1999 Lemma 2.2]. We emulate the same process using the character table in place of the dual group of our underlying group. We write down a convenient $ L $-basis of $L[N]$ and form $ G $-invariant linear combinations of these basis elements. We find that certain quaternion algebras appear in the decompositions, and so we fix notation for these: for $ x,y \in F^{\times} $, let $ (x,y)_{F} $ denote the quaternion algebra with $ F $-basis $ 1,u,v,w $ satisfying the relations $ u^{2} = x $, $ v^{2} = y $, and $ uv = w = -vu $. In addition, let $ a= \alpha^{2} \in F^{\times} $, $ b=\beta^{2} \in F^{\times} $, where $ \alpha, \beta \in K $ are as defined at the beginning of this section.

We begin with the Hopf algebras giving the classical and canonical non-classical structures of type $Q_8$.

