d
  \( \tau \phi_3 \tau^{-1} \) fix the elements of \( \tau(M) \), they
  are both \( \tau(M) \)-automorphisms of \( L \).  Therefore \( \tau
  M^* \tau^{-1} \subseteq \tau(M)^{*} \).
</p>
<h3 id="lhs-superset-of-rhs">LHS &supe; RHS</h3>
<p>
  Consider the set \( \tau^{-1} \tau(M)^* \tau \) and examine how its
  elements transform the elements of \( M \).  For all \( k + l
  \sqrt{2} \in M \), we get

  \begin{align*}
    (\tau^{-1} \phi_1 \tau)(k + l \sqrt{2})
    &= (\tau^{-1} \phi_1)(k - \sqrt{2}) \\
   