t we find

  \begin{equation}
    \tau(M)^* = \{ \phi_1, \phi_3 \}.
    \label{eq-tau-m-ast}
  \end{equation}

  Now

  \begin{align*}
    \tau M^* \tau^{-1}
    &= \{ \tau \gamma \tau^{-1} : \gamma \in {M^*} \} \\
    &= \{ \tau \phi_1 \tau^{-1}, \tau \phi_3 \tau^{-1} \}.
  \end{align*}

  Let us now find out how each element of \( \tau M^* \tau^{-1} \)
  transforms the elements of \( L \).  For all \( a + b \sqrt{2} + c
  \sqrt{3} + d \sqrt{6} \in L \), we get

  \begin{align*}
    (\tau \phi_1 \tau^{-1})