  We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann
subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for
$\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof to handle
$\textbf{k}=\mathbb{Z}$. The key step for the proof is a complete description
of all intermediate von Neumann subalgebras between $L(SL_2(\textbf{k}))$ and
$L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where
$SL_2(\textbf{k})\curvearrowright Y$ denotes the quotient of the algebraic
action $SL_2(\textbf{k})\curvearrowright \widehat{\textbf{k}^2}$ by modding out
the relation $\phi\sim \phi'$, where $\phi$, $\phi'\in \widehat{\textbf{k}^2}$
and $\phi'(x, y):=\phi(-x, -y)$ for all $(x, y)\in \textbf{k}^2$. As a
by-product, we show $L(PSL_2(\mathbb{Q}))$ is a maximal von Neumann subalgebra
in $L^{\infty}(Y)\rtimes PSL_2(\mathbb{Q})$; in particular,
$PSL_2(\mathbb{Q})\curvearrowright Y$ is a prime action, i.e. it admits no
non-trivial quotient actions.
