\label{eq:5-25}$$ where ${}_{\pm 2}I_{(lm)(lm)'}$ are the Hermitian coupling matrices $${}_{\pm 2} I_{(lm)(lm)'} = \int {{\rmn{d}}}{\hat{\bmath{n}}}\, w({\hat{\bmath{n}}})
{}_{\pm 2} Y_{(lm)'}({\hat{\bmath{n}}}){}_{\pm 2} Y_{lm}^* ({\hat{\bmath{n}}}).
\label{eq:4}$$ Using ${}_s Y_{lm}^* = (-1)^{s+m} {}_{-s}Y_{l-m}$, and the reality of $w({\hat{\bmath{n}}})$, we have ${}_{\pm 2} I_{(lm)(lm)'}^* = (-1)^{m+m'}
{}_{\mp 2} I_{(l-m)(l-m)'}$ which ensures that $\tilde{E}_{lm}^* =
(-1)^m \tilde{E}_{l-m}$ and a similar result for $\tilde{B}_{lm}$. If we expand the weight function in terms of its spherical multipoles, $w({\hat{\bmath{n}}}) = \sum_{lm} w_{lm} Y_{lm}({\hat{\bmath{n}}})$, the coupling matrices can be evaluated in terms of Wigner-$3j$ symbols: $${}_{\pm 2} I_{(lm)(lm)'} = \sum_{LM} (-1)^m w_{LM}
\sqrt{\frac{(2L+1)(2l+1)(2l'+1)}{4\pi}} 
\left( \begin{array}{ccc} l & l' & L \\ m & -m' & -M \end{array} \right) 
\left( \begin{array}{ccc} l & l' & L \\ \mp 2&\pm 2 &0 \end{array} \right). 
\label{eq:5-26}$$ From equation (\[eq:5-25\]) we can separate the $E$ and $B$ pseudo-multipoles as $$\begin{aligned}
\tilde{E}_{lm} &=& \sum_{(lm)'} 
\left( {}_+ I_{lm(lm)'}E_{(lm)'} + i{}_{-}I_{(lm)(lm)'}B_{(lm)'} \right),
\label{eq:5-27}\\
\tilde{B}_{lm} &=& \sum_{(lm)'}
\left( {}_+ I_{(lm)(lm)'}B_{(lm)'} - i{}_{-}I_{(lm)(lm)'}E_{(lm)'} \right),
\label{eq:5-28}\end{aligned}$$ where we define the Hermitian ${}_\pm I_{(lm)(lm)'}$ to be $${}_+ I_{(lm)(lm)'} \equiv \frac{1}{2}({}_{+2} I_{(lm)(lm)'}
+ {}_{-2}I_{(lm)(lm)'}),
\qquad {}_{-}I_{(lm)(lm)'} \equiv \frac{1}{2}({}_{+2}I_{(lm)(lm)'} -
{}_{-2}I_{lm(lm)'}).
\label{eq:5-29}$$ Given the $\tilde{E}_{lm}$ and $\tilde{B}_{lm}$, we form their pseudo-$C_l$s according to $$\tilde{C}_l^E \equiv \frac{1}{2l+1}\sum_{m} |\tilde{E}_{lm}|^2, \quad
\tilde{C}_l^B \equiv \frac{1}{2l+1}\sum_{m} |\tilde{B}_{lm}|^2.
\label{eq:9}$$ 