\\
 & = & \sum_{r>0,r\in\mathbb{Z}+\frac{1}{2}}\left(r^{2}+\frac{3}{4}\right)(\psi_{-r}\psi_{r}^{*}-\psi_{-r}^{*}\psi_{r})\,.\nonumber\end{aligned}$$

Fermionization of triple bosons term
------------------------------------

Now let us consider the term $$\begin{aligned}
 &  & (1-b^{2})(-\sum_{n,m>0}\tilde{a}_{-n}\tilde{a}_{-m}\tilde{a}_{n+m})\\
 &  & =\frac{2}{3}(1-b^{2})\sum_{n}[-2\tilde{a}_{-n}\tilde{L}_{n}+\tilde{L}_{-n}\tilde{a}_{n}]\\
 &  & =\frac{2}{3}(1-b^{2})\left(\sum_{n}{{\diamond}\atop {\diamond}}\tilde{L}_{-n}\tilde{a}_{n}-2\tilde{a}_{-n}\tilde{L}_{n}{{\diamond}\atop {\diamond}}+{\text{Contractions}}\right)\,.\end{aligned}$$ In formulation of $\tilde{a}$ and $\tilde{L}$, it is easier to calculate the fermionic expression. We have $$\begin{aligned}
{{\diamond}\atop {\diamond}}\tilde{L}_{-n}\tilde{a}_{n}-2\tilde{a}_{-n}\tilde{L}_{n}{{\diamond}\atop {\diamond}} & = & \sum_{\begin{subarray}{c}
r\in\mathbb{Z}+\frac{1}{2}\\
s\in\mathbb{Z}+\frac{1}{2}
\end{subarray}}\left((r-\frac{n}{2}):\psi_{-r}\psi_{-n+r}^{*}\psi_{-s}\psi_{n+s}^{*}:\right.\\
 &  & -\left.(2s+n):\psi_{-r}\psi_{-n+r}^{*}\psi_{-s}\psi_{n+s}:\right)\nonumber \\
 & = & \sum_{\begin{subarray}{c}
r,s,k,l\in\mathbb{Z}+\frac{1}{2}\\
r+s<0,l+k>0\\
r+s+k+l=0
\end{subarray}}:\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}:\left(\frac{1}{2}(s-r)+k-l\right)\,.\nonumber\end{aligned}$$ To calculate the contractions, we need to calculate the commutator $$\begin{aligned}
[\tilde{L}_{-n},\psi_{-k}] & = & \sum_{r}\left[(r-\frac{n}{2}):\psi_{-r}\psi_{-n+r}^{*}:\,,\,\,\psi_{-k}\right]\\
 & = & \left(\theta(k>0)\sum_{r>0}\delta_{r-n,k}\psi_{-r}\right.\nonumber \\
 &  & +\left.\theta(k<0)\sum_{r<0}\delta_{r-n,k}\psi_{-r}\right)(r-\frac{n}{2})\nonumber \\
 & = & (k+\frac{n}{2})\psi_{-n-k}\nonumber \\
\,[\tilde{L}_{-n},\psi_{n+k}^{*}] & = & \sum_{r}\left[(r-\frac{n}{2}):\psi_{-r}\psi_{-n+r}^{*}:\,,\,\,\psi_{n+k}^{*}\right]\\
 & = & \left(-\theta(n+k>0)\sum_{r>0}\delta_{r,n+k}\psi_{-n+r}^{*}\right.\nonumber \\
 &  & -\left.\theta(n+k<0)\sum_{r<0}\delta_{r,n+k}\psi_{-n+r}^{*}\right)(r-\frac{n}{2})\nonumber \\
 & = & -(k+\frac{n}{2})\psi_{k}^{*}\nonumber \\
\,[\psi_{-k},\tilde{L}_{n}] & = & (-k+\frac{n}{2})\psi_{n-k}\\
\,[\psi_{-n+k}^{*},\tilde{L}_{n}] & = & (k-\frac{n}{2})\psi_{k}^{*}\,.\end{aligned}$$ Hence we have the contraction as follows $$\begin{aligned}
{\text{Contractions}} & = & \left(\sum_{n\in\mathbb{Z},k>0}:[\tilde{L}_{-n},\psi_{-k}]\psi_{n+k}^{*}:+\sum_{n\in\mathbb{Z},n+k<0}:\psi_{-k}[\tilde{L}_{-n},\,\psi_{n+k}^{*}]:\right.\nonumber \\
 &  & -2\left.\left(\sum_{n\in\mathbb{Z},n-k<0}:\psi_{-k}[\psi_{-n+k}^{*},\tilde{L}_{n}]:+\sum_{n\in\mathbb{Z},k<0}:[\psi_{-k},\,\tilde{L}_{n}]\psi_{-n+k}^{*}:\right)\right)\nonumber \\
 & = & \left(\sum_{k>0}(k+\frac{n}{2}):\psi_{-n-k}\psi_{n+k}^{*}:-\sum_{n+k<0}(k+\frac{n}{2}):\psi_{-k}\psi_{k}^{*}:\right.\nonumber \\
 &  & -2\left.\sum_{n<k}(k-\frac{n}{2}):\psi_{-k}\psi_{k}^{*}:+2\sum_{k<0}(k-\frac{n}{2}):\psi_{n-k}\psi_{-n+k}^{*}:\right)\nonumber \\
 & = & \sum_{k>n}(k-\frac{n}{2}):\psi_{-k}\psi_{k}:-\sum_{k<-n}(k+\frac{n}{2}):\psi_{-k}\psi_{k}^{*}:\nonumber \\
 &  & +\sum_{k<-n}(2k+n):\psi_{-k}\psi_{k}^{*}:-\sum_{k>n}(2k-n):\psi_{-k}\psi_{k}^{*}:\nonumber \\
 & = & \left(\sum_{k>n}(\frac{n}{2}-k)+\sum_{k<-n}(k+\frac{n}{2})\right):\psi_{-k}\psi_{k}^{*}:\nonumber \\
 & = & \sum_{0<n<k}(\frac{n}{2}-k)(\psi_{-k}\psi_{k}^{*}-\psi_{-k}^{*}\psi_{k})\\
 & = & \sum_{k>0}\left((-k)(k-\frac{1}{2})+\frac{1}{4}(k+\frac{1}{2})(k-\frac{1}{2})\right)(\psi_{-k}\psi_{k}^{*}-\psi_{-k}^{*}\psi_{k})\nonumber \\
 & = & -\frac{1}{16}\sum_{k>0,k\in\mathbb{Z}+\frac{1}{2}}(6k-1)(2k-1)(\psi_{-k}\psi_{k}^{*}-\psi_{-k}^{*}\psi_{k})\end{aligned}$$ Now the whole $H_{1}$ is $$\begin{aligned}
H_{1} & = & \frac{2}{3}(1-b^{2})\left(\sum_{\begin{subarray}{c}
r,s,k,l\in\mathbb{Z}+\frac{1}{2}\\
r+s<0,l+k>0\\
r+s+k+l=0
\end{subarray}}:\psi_{r}\psi_{s}^{*}\psi_{k}\psi_{l}^{*}:\left(\frac{1}{2}(s-r)+k-l\right)\right.\\
 &  & -\left.\frac{1}{16}\sum_{k>0,k\in\mathbb{Z}+\frac{1}{2}}(6k-1)(2k-1)(\psi_{-k}\psi_{k}^{*}-\psi_{-k}^{*}\psi_{k})\right)\nonumber\end{aligned}$$

Fermionization 