(a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6})
    &= (\tau \phi_1)(a - b\sqrt{2} + c\sqrt{3} - d\sqrt{6}) \\
    &= \tau (a - b\sqrt{2} + c\sqrt{3} - d\sqrt{6}) \\
    &= a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}).
  \end{align*}

  Therefore

  \[
    \tau \phi_1 \tau^{-1} = \phi_1.
  \]

  Similarly,

  \begin{align*}
    (\tau \phi_3 \tau^{-1})(a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6})
    &= (\tau \phi_3)(a - b\sqrt{2} + c\sqrt{3} - d\sqrt{6}) \\
    &= \tau (a - b\sqrt{2} - c\sqrt{3} + d\sqrt{6}) \\
    &= a 