symbol may be used for quantities that differ by signs, factors of two, or complex conjugation. Here we follow the definitions of Davidson and Haber [@DavHab]. With these definitions: $$\begin{aligned}
Y_{11}=m_{11}^2, &&
Y_{12}=-m_{12}^2,
\nonumber \\
Y_{21}=-(m_{12}^2)^\ast && Y_{22}=m_{22}^2,
\label{ynum}\end{aligned}$$ and $$\begin{aligned}
Z_{11,11}=\lambda_1, && Z_{22,22}=\lambda_2,
\nonumber\\
Z_{11,22}=Z_{22,11}=\lambda_3, && Z_{12,21}=Z_{21,12}=\lambda_4,
\nonumber \\
Z_{12,12}=\lambda_5, && Z_{21,21}=\lambda_5^\ast,
\nonumber\\
Z_{11,12}=Z_{12,11}=\lambda_6, && Z_{11,21}=Z_{21,11}=\lambda_6^\ast,
\nonumber \\
Z_{22,12}=Z_{12,22}=\lambda_7, && Z_{22,21}=Z_{21,22}=\lambda_7^\ast.
\label{znum}\end{aligned}$$

Basis transformations
---------------------

The scalar potential can be rewritten in terms of new fields $\Phi^\prime_a$, obtained from the original ones by a simple (global) basis transformation $$\Phi_a \rightarrow \Phi_a^\prime = U_{ab} \Phi_b,
\label{basis-transf}$$ where $U\in U(2)$ is a $2 \times 2$ unitary matrix. Under this unitary basis transformation, the gauge-kinetic terms are unchanged, but the coefficients $Y_{ab}$ and $Z_{ab,cd}$ are transformed as $$\begin{aligned}
Y_{ab} & \rightarrow &
Y^\prime_{ab} =
U_{a \alpha}\, Y_{\alpha \beta}\, U_{b \beta}^\ast ,
\label{Y-transf}
\\
Z_{ab,cd} & \rightarrow &
Z^\prime_{ab,cd} =
U_{a\alpha}\, U_{c \gamma}\,
Z_{\alpha \beta,\gamma \delta}\, U_{b \beta}^\ast \, U_{d \delta}^\ast .
\label{Z-transf}\end{aligned}$$ Thus, the basis transformations $U$ may be utilized in order to absorb some of the degrees of freedom of $Y$ and/or $Z$, which implies that not all parameters of Eq. (\[VH2\]) have physical significance.

\[subsec:symmetries\]The six classes of symmetries in 