The Kronecker Hall algebra strictly contains its composition subalgebra

The acyclic Euler form is $\chi (\alpha ,\beta )={\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2-2{\alpha }_1{\beta }_2$, which is bilinear by [F1]. Define this example’s product by $[M]\star [N]={\nu }^{\chi (\mathrm{deg}M,\mathrm{deg}N)}[M]\ast [N]$, where $\ast$ is the untwisted Hall product of [F3]. This twist preserves associativity, since $\chi (\alpha ,\beta )+\chi (\alpha +\beta ,\gamma )=\chi (\beta ,\gamma )+\chi (\alpha ,\beta +\gamma )$. Thus the composition subalgebra is defined for this quiver.

A representation of dimension $(1,1)$ is a pair $(x,y)\in {\mathbb{F}}_q^2$ of arrow scalars. An isomorphism is a pair of nonzero endpoint scalars $(g_1,g_2)$, acting by common multiplication by $g_2\mathrm{/}{g}_1$. The zero pair is one orbit and gives $S_1\oplus S_2$. Each nonzero orbit is a line in ${\mathbb{F}}_q^2$, giving $q+1$ further orbits. Thus the full Hall component has $q+2$ independent isomorphism-class basis vectors.

A product of simples of total degree $(1,1)$ must use $S_1$ and $S_2$ once each. There are only two possible words, $[S_1]\star [S_2]$ and $[S_2]\star [S_1]$, so the degree component generated by simples has dimension at most two.

Since $q\ge 2$ by [F4], $q+2>2$. The two subspaces have different dimensions, proving strict containment.