n*}

  Similarly, for all \( k - l \sqrt{2} \in \tau(M) \), we get

  \begin{align*}
    (\tau \phi_3 \tau^{-1})(k - l \sqrt{2})
    &= (\tau \phi_3)(k + l \sqrt{2}) \\
    &= \tau(k + l \sqrt{2}) \\
    &= k - l \sqrt{2}.
  \end{align*}

  Note above that both \( \phi_1 \) and \( \phi_3 \) fix \( k + l
  \sqrt{2} \in M \) because \( \phi_1, \phi_2 \in M^* \), the set of
  \( M \)-automorphisms of \( L \).  This detail will be used in the
  general proof.
</p>
<p>
  Since both \( \tau \phi_1 \tau^{-1} \) an