In the [paper][1] by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana wires which in $\mathfrak{so}(2n)$, $n=2$. Then based on the fact that $\mathfrak{so(4)}\cong\mathfrak{so(3)}\oplus\mathfrak{so(3)}$, they aim to get the generators of two copies of $\mathfrak{so(3)}$ (which I do not understand why they used two copies of $\mathfrak{so(3)}$ rather than just one $\mathfrak{so(4)}$?). The map of Lie algebra is defined:

\begin{align}

\rho(A)=\frac{i}{4} \sum\limits_{j,k=1}^{2n} A_{jk}\hat{c}_j\hat{c}_k

\end{align}

Where $A \in \mathfrak{so}(2n)$ is an anti-symmetric matrix belonging to the Lie algebra. So here we have $n=2$ and 4 Majoranas $\{\hat{c}_1,\hat{c}_2,\hat{c}_3,\hat{c}_4\}$.

I have three questions:

1. Here $\rho$ is the adjoint representation? Base on the relation

\begin{align}

[-i\rho(A),-i\rho(B)]=-i\rho([A,B])

\end{align}

Therefore $\rho$ defines a Lie algebra, so is it the adjoint representation? Also, I am not sure why I am not able to finish the proof of the above equation. Here is my attempt (I know I got the commutation of $A$ and $B$ but I do not know how to get the other direction:

\begin{align}

[-i\rho(A),-i\rho(B)]&=-i(\frac{i}{4})^2 \sum\limits_{j,k=1}^{2n} \sum\limits_{j',k'1}^{2n'}(A_{jk}\hat{c}_j\hat{c}_k B_{j'k'}\hat{c}_{j'}\hat{c}_{k'}-B_{j'k'}\hat{c}_{j'}\hat{c}_{k'}A_{jk}\hat{c}_j\hat{c}_k)\\

&=-i(\frac{i}{4})^2 \sum\limits_{j,k=1}^{2n} \sum\limits_{j',k'1}^{2n'}(A_{jk} B_{j'k'}-B_{j'k'}A_{jk})\hat{c}_j\hat{c}_k\hat{c}_{j'}\hat{c}_{k'}\\

&=...

\end{align}

2. How the induced action on $\hat{c}$ is defined as follows?

\begin{align}

\hat{c}_l \rightarrow i[\rho(A),c_l]

\end{align}

I cannot understand this induced action? and they did not even define $c_l$.

3. My big question is that how under $\rho$ the generators of two $\mathfrak{so(3)}$ obey equations (5) and (6) in the reference. I know $\mathfrak{so(4)}$ has 6 generators.

\begin{align}

&A_1=\left[\begin{array}{cccc}

0 & 0 & 0 & 0 \\

0 & 0& -1 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 0 & 0

\end{array}\right] \quad A_2=\left[\begin{array}{cccc}

0 & 0 & 1 & 0 \\

0 & 0& 0& 0 \\

-1 & 0 & 0 & 0 \\

0 & 0 & 0 & 0

\end{array}\right] \quad A_3=\left[\begin{array}{cccc}

0 & -1 & 0 & 0 \\

1 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0

\end{array}\right] \\

&B_1=\left[\begin{array}{cccc}

0 & 0 & 0 & -1 \\

0 & 0& 0& 0 \\

0 & 1 & 0 & 0 \\

1 & 0 & 0 & 0

\end{array}\right] \quad B_2=\left[\begin{array}{cccc}

0 & 0 & 0 & 0 \\

0 & 0& 0& -1 \\

0 & 0 & 0 & 0 \\

0 & 1 & 0 & 0

\end{array}\right] \quad B_3=\left[\begin{array}{cccc}

0 & 0& 0 & 0 \\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & -1 \\

0 & 0 & 1 & 0

\end{array}\right] \\

\end{align}

So as the paper addressed both $A_i$ and $B_i$ belong to $\mathfrak{so(4)}$, but I don't understand how to get the generators.

[1]: https://arxiv.org/abs/0904.2197