simultaneously impose $\Pi_2$ and $U(1)$ *in the same basis*, then one easily sees that the conditions of are satisfied. The same conclusion also follows if we simultaneously impose $U(1)$ and CP2 in the same basis. Moreover, starting from a scalar potential in which the conditions of are satisfied, a basis transformation can be found (see Ref. [@FHS] for the details) such that $\lambda_5=\lambda_1-\lambda_3-\lambda_4$ in the new basis, as required in the CP3-symmetric THDM \[cf. Table \[master1\]\].

Finally, we indicate the basis-independent conditions that guarantee that a basis exists in which the CP3-symmetric conditions are satisfied. Defining $Z_{ab}^{(1)}\equiv Z_{ac,cb}$, we first require that [@DavHab] \[erps\] Y\_[ab]{}=m\_[11]{}\^2\_[ab]{} \[Z\^[(1)]{}\]\^2=([Tr]{} Z\^[(1)]{})\^2. If is satisfied, then \[erps2\] m\_[11]{}\^2=m\_[22]{}\^2,m\_[12]{}\^2=0,\_1=\_2,\_7=-\_6, must hold in *all basis choices*. This is the so-called exceptional region of parameter space (ERPS) of the THDM. In Ref. [@FHS], we then constructed a second invariant quantity $D$ built out of the $Y_{ab}$ and $Z_{ab,cd}$ with the following property: if $D=0$ in the ERPS, then there exists a basis for the scalar fields such that the CP3-conditions of Table \[master1\] are satisfied.[^6] Thus, the conditions for the ERPS plus $D=0$ provide necessary (although not sufficient) invariant conditions that must be satisfied by the MSSM Higgs 