U X U^\top.
\label{X-prime}\end{aligned}$$ Hence, a basis-invariant treatment is critical for distinguishing between two potentially different symmetries.

Of course, one may combine several HF symmetries and/or GCP symmetries. Ivanov [@Ivanov1] has proved that, whatever combination one chooses, one will end up in one of six distinct classes of symmetries. In a recent article we have clarified this issue showing how to construct such classes with simple examples [@FHS]. The result is shown in Table \[master1\].

   symmetry   $m_{11}^2$   $m_{22}^2$    $m_{12}^2$   $\lambda_1$   $\lambda_2$   $\lambda_3$         $\lambda_4$                        $\lambda_5$                   $\lambda_6$    $\lambda_7$
  ---------- ------------ ------------- ------------ ------------- ------------- ------------- ------------------------- -------------------------------------------- ------------- ---------------
    $Z_2$                                    0                                                                                                                              0              0
    $U(1)$                                   0                                                                                                0                             0              0
   $SO(3)$                 $ m_{11}^2$       0                      $\lambda_1$                 $\lambda_1 - \lambda_3$                       0                             0              0
     CP1                                    real                                                                                             real                         real           real
     CP2                   $m_{11}^2$        0                      $\lambda_1$                                                                                                      $- \lambda_6$
     CP3                   $m_{11}^2$        0                      $\lambda_1$                                           $\lambda_1 - \lambda_3 - \lambda_4$ (real)        0              0

  : Impact of the symmetries on the coefficients of the Higgs potential in a specified basis. See Ref. [@FHS] for more details.

\[master1\]

Five of the symmetry classes may be imposed by the following single requirements: $$\begin{aligned}
Z_2: &\hspace{1cm} &
S =
\left(
\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}
\right),
\\
U(1): &\hspace{1cm} &
S =
\left(
\begin{array}{cc}
