= g^{\prime\,2}\,,
\nonumber \\
f_U^a f_U^{\prime\,a \ast} = gg^\prime,
&\hspace{1cm}&
f_D^{a \ast} f_D^{\prime a} = gg^{\prime}\,.
\label{f_ESL}\end{aligned}$$

To establish U(2)-invariant conditions that enforce a supersymmetric scalar Higgs potential, we first construct basis-independent quantities that involve both the scalar potential parameters and the gaugino-higgsino-Higgs couplings. For example,[^9] $$\begin{aligned}
{\cal Y}_{DD} &=& \hat f_D^a\, \hat f_D^{b \ast}\ Y_{ab},
\qquad\qquad
{\cal Y}_{UU} = \hat f_U^a\, \hat f_U^{b \ast}\ Y_{ab},
\nonumber\\
{\cal Y}_{DU} &=& \hat f_D^a\, \hat f_U^{b \ast}\ Y_{ab},
\qquad\qquad
{\cal Y}_{UD} = \hat f_U^a\, \hat f_D^{b \ast}\ Y_{ab},
\label{calY}\end{aligned}$$ provide basis-invariant quantities involving the quadratic coefficients of the Higgs potential. Likewise, $${\cal Z}_{\alpha \beta, \gamma \delta} =
\hat f_\alpha^a\, \hat f_\beta^{b \ast}\,
\hat f_\gamma^c\, \hat f_\delta^{c \ast}\, Z_{ab,cd}\,,
\label{calZ}$$ where the indices $\alpha$, $\beta$, $\gamma$, and $\delta$ can take the values $D$ or $U$, provide basis-invariant quartic coefficients for the Higgs potential.

Evaluating the invariant quantities introduced in in the supersymmetric basis defined by Eqs. (\[yESL\]), (\[zESL\]) and (\[spbasis\]), it follows that \_[DD]{} = [Y]{}\_[UU]{},\_[DU]{} = [Y]{}\_[UD]{} = 0, \[calY\_ESL\] and $$\begin{aligned}
{\cal Z}_{DD,DD}&=& {\cal Z}_{UU,UU}  = \tfrac{1}{4}\left[ f_U^{a\,\ast} f_U^a+
 f_D^{a\,\ast} f_D^a\right],
\nonumber\\
{\cal Z}_{DD,UU} &=& {\cal Z}_{UU,DD} = \tfrac{1}{4}
\left[ f_U^{a\,\ast} f_U^a - f_U^{\prime\,a\,\ast} f_U^{\prime\,a}\right],
\nonumber\\
{\cal Z}_{DU,UD} &=& {\cal Z}_{UD,DU} 