we consider contributions to the Gibbs free energy specific for non-chiral and chiral superconductors respectively.

Non-chiral Triplet Superconductor
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As mentioned earlier, here we assume the orbital dependence of the gap function to be the same for all spin-triplet components. Formally, this means that the orbital degree of freedom belongs to a one-dimensional irreducible representation of the point group. We apply our analysis to materials that have orthorhombic, tetragonal, or hexagonal point groups. One relevant example is a non-chiral triplet $f$-wave superconductor with hexagonal symmetry, which has been proposed in the context of the the cobaltates Na$_x$CoO$_2\cdot y$H$_2$O; in this case $f({\bf k}) = k_x(k_x^2-3k_y^2)$ from Eq.. When the ${\bf d}$-vector lies in the xy plane, the inclusion of spin-orbit coupling implies that formally this order parameter belongs to the $\Gamma_6^-$ representation of the hexagonal point group (the consequences of spin-orbit coupling for this representation is discussed in more detail in Section \[sec:SO\]).

With the in-plane spin rotational invariance, the relevant free energy within the assumptions stated above is given by $$\begin{aligned}
&f^{(2)}_{in} = \sum_{i=x,y,z} K_i(|D_i\Delta_{\uparrow\uparrow}|^2+
|D_i\Delta_{\downarrow\!\downarrow}|^2),\label{EQ:f2in-NX}\\
&f^{(4)}_{hom}=\beta_1(\sum_{i}|\Delta_{i}|^2)^2+\beta_2|\Delta_{\uparrow\!\uparrow}|^2
|\Delta_{\downarrow\!\downarrow}|^2,\label{EQ:f2hom-NX}\\
&f^{(4)}_{in} =
\gamma[\Delta_{\uparrow\!\uparrow}^*\Delta_{\downarrow\!\downarrow}(\textbf{D}_{\perp}\Delta_{\uparrow\!\uparrow})\!\cdot\!(\textbf{D}_{\perp}\Delta_{\downarrow\!\downarrow})^*+{\rm c.c}],\label{EQ:f4-NX}\end{aligned}$$ For tetragonal and hexagonal point groups $K_x=K_y$ while for orthorhombic point groups, $K_x\ne K_y$. For the high 