  Once again, let us see how each element of \( \tau M^* \tau^{-1} \)
  transforms the elements of \( \tau(M) \).  Note that this time we
  are not going to examine how they transform arbitrary elements of \(
  L \).  We are only going to see how they transform the elements of
  \( \tau(M) \).  For all \( k - l \sqrt{2} \in \tau(M) \), we get

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