diff --git a/main.tex b/main.tex
index 214c5f1901e97652a7a66a1c8bedec1877e56d93..a3f0ca613e1e35c1c5a3863bb17451b8fa53183a 100644
--- a/main.tex
+++ b/main.tex
@@ -5,6 +5,7 @@ superscriptaddress, floatfix, longbibliography]{revtex4-2}
 
 \usepackage[version=4]{mhchem}
 \usepackage[dvipsnames]{xcolor}
+\usepackage{bbold}
 
 \definecolor{first}{RGB}{0,153,230}
 \definecolor{second}{RGB}{40,37,110}
@@ -150,6 +151,13 @@ studies\,\cite{wolf2018,wolf2022a}, we treat the interacting part
 $H_\mathrm{int}$ with the functional renormalization group (FRG), a short
 introduction is given below.
 
+Previous FRG studies including Rashba-SOC have been conducted on the
+square-lattice Rashba Hubbard model in Ref.~\onlinecite{beyer2023}, on the
+triangular lattice with attractive electron electron interactions in
+Ref.~\onlinecite{schober2016}.
+Studies for SOC of non-Rashba type have been conducted on twisted bilayer
+\ce{PtSe2} in Ref.~\onlinecite{klebl2022a}.
+
 \section{Model and Method}
 \label{sec:modelmethod}
 
@@ -214,7 +222,6 @@ neighbor density-density interaction according to
 \end{equation}
 where $\bar \sigma$ is the opposing spin to $\sigma$.
 
-
 \subsection{Truncated Unity Functional Renormalization Group}
 In order to properly resolve the different possible particle-particle and
 particle-hole instabilities of the triangular lattice Rashba-Hubbard model it
@@ -224,12 +231,6 @@ we choose the functional renormalization group (FRG) for its wide applicability
 and advantageous numerical scaling.
 Furthermore, the results of FRG can trivially exhibit singlet-triplet mixing,
 which we expect to find in SOC systems.
-Previous FRG studies including Rashba-SOC have been conducted on the
-square-lattice Rashba Hubbard model in Ref.~\onlinecite{beyer2023}, on the
-triangular lattice with attractive electron electron interactions in
-Ref.~\onlinecite{schober2016}.
-Studies for SOC of non-Rashba type have been conducted on twisted bilayer
-\ce{PtSe2} in Ref.~\onlinecite{klebl2022a}.
 
 \begin{figure}
     \centering
@@ -273,14 +274,16 @@ Ref.\,\onlinecite{beyer2022a}.
 \subsection{Analysis of Results}
 
 \subsubsection{Particle-particle instabilities}
-To accurately determine the nature of particle-particle instabilities we
-calculate the truncated unity susceptibility as previously described in
+To determine the nature of particle-particle instabilities we calculate the
+P-channel truncated unity susceptibility as previously described in
 Ref.~\cite{beyer2022a,klebl2022a}.
 The obtained eigenstate we decompose into singlet-like ($\Psi$) and
 triplet-like ($d_x, d_y, d_z$) components, calculating the magnitude the
-respective subspaces.
-\todo{I think the d basis should be defined explicitly here, 
-heavily used after this point.}
+respective subspaces according to
+\begin{equation}
+    v = [\Psi(\bvec k) \mathbb{1} + \bvec d(\bvec k)
+        \cdot \hat{\bvec{\sigma}}] i \hat \sigma_y \, .
+\end{equation}
 Notably, at finite Rashba-SOC the eigenstate is expected to have finite weight
 in both subspaces, due to the singlet-triplet mixing.
 
@@ -610,21 +613,28 @@ effect of the Rashba SOC.
 \section{Irreducible representation basis functions in the spin representation}
 \label{app:spin-irreps}
 
-To get the irreducible representations of the two spins, we begin by constructing 
-the symmetry group representation of for a single spin, which we denote as $S(C_{6v})$. 
-The representation of the whole group can be constructed from the two generators,
-which are one of the $C_6$ rotations:
+To get the irreducible representations of the two spins, we begin by
+constructing the symmetry group representation of for a single spin, which we
+denote as $S(C_{6v})$.  The representation of the whole group can be
+constructed from the two generators, which are one of the $C_6$ rotations:
 \begin{equation}
-    S(C_6) = \exp \bigg( \frac{-i \pi \hat{\tau}_z}{6} \bigg) =  
+    S(C_6) = \exp \bigg( \frac{-i \pi \hat{\sigma}_z}{6} \bigg)
+    =
     \begin{pmatrix}
         \sqrt{3} - i & 0 \\
-        0 & \sqrt{3} + i 
+        0 & \sqrt{3} + i
     \end{pmatrix} ,
 \end{equation}
-where $\tau_z$ is the third Pauli matrix, as well as a reflection $\sigma_1$,
-which we will take to be the reflection in the $y$-axis:
+where $\hat\sigma_z$ is the third Pauli matrix, as well as a reflection $m_y$,
+which we will take to be the mirror w.r.t. the $y$-axis:
 \begin{equation}
-    S(\sigma_1) = i \hat{\tau}_x \, .
+    S(m_y) = i \hat{\sigma}_x
+    =
+    \begin{pmatrix}
+        0 & i \\
+        i & 0
+    \end{pmatrix}
+    \, .
 \end{equation}
 
 Given the symmetry group representation acting on a single spin $S(C_{6v})$, the