case given in other literature, e.g., Morisawa and Ida [@MI], and therefore we present here only some key formulas. Following  [@BCCGSW], we introduce the $G_{2(2)}/SO(4)$ coset matrix, $M$, defined by $$\begin{aligned}
M= \left(
  \begin{array}{ccc}
  \hat A&\hat B&\sqrt{2}\hat U\\
  \hat B^T&\hat C&\sqrt{2}\hat V\\
  \sqrt{2}\hat U^T&\sqrt{2}\hat V^T&\hat S\\
  \end{array}
 \right) \,,\end{aligned}$$ where $\hat A$ and $\hat C$ are symmetric $3\times 3$ matrices, $\hat B$ is a $3\times 3$ matrix, $\hat U$ and $\hat V$ are 3-component column matrices, and $\hat S$ is a scalar, defined, respectively, by $$\begin{aligned}
&&\hat A=\left(
  \begin{array}{ccc}
  [(1-y)\lambda+(2+x)\psi \psi^T-\tau^{-1}\tilde\omega\tilde\omega^T+\mu(\psi \psi^T\lambda^{-1}\hat J-\hat J\lambda^{-1}\psi\psi^T)]&\tau^{-1}\tilde\omega\\
 \tau^{-1}\tilde\omega^T& -\tau^{-1}
  \end{array}
 \right) \,,\nonumber\\
&&\hat B=\left(
  \begin{array}{ccc}
  (\psi\psi^T-\mu \hat J)\lambda^{-1}-\tau^{-1}\tilde\omega \psi^T \hat J&[(-(1+y)\lambda \hat J-(2+x)\mu+\psi^T\lambda^{-1}\tilde\omega)\psi+(z-\mu \hat J\lambda^{-1}\tilde)\omega] \\
  \tau^{-1}\psi^T \hat J&-z\\
  \end{array}
 \right) \,, \nonumber\\
&&\hat C=\left(
  \begin{array}{ccc}
 (1+x)\lambda^{-1}-\lambda^{-1}\psi\psi^T\lambda^{-1}&\lambda^{-1}\tilde\omega-\hat J(z-\mu \hat J \lambda^{-1})\psi\\
 \tilde\omega^T\lambda^{-1}+\psi^T(z+\mu \lambda^{-1}\hat J)\hat J&[\tilde\omega^T\lambda^{-1}\tilde\omega-2\mu\psi^T\lambda^{-1}\tilde\omega-\tau(1+x-2y-xy+z^2)]\\
  \end{array}
 \right) \,,\nonumber\\
&&\hat U=
\left(
  \begin{array}{cc}
  (1+x-\mu \hat J\lambda^{-1})\psi-\mu \tau^{-1}\tilde\omega\\
  \mu\tau^{-1}\\
  \end{array}
 \right) \,,\nonumber\\
&&\hat V=\left(
  \begin{array}{ccc}
  (\lambda^{-1}+\mu\tau^{-1}\hat J)\psi\\
  \psi^T\lambda^{-1}\tilde\omega-\mu(1+x-z)\nonumber\\
  \end{array}
 \right) \,,\nonumber\\
&&\hat S=1+2(x-y) \,, \nonumber\end{aligned}$$ with $$\begin{aligned}
&&\tilde \omega=\omega-\mu\psi \,,\end{aligned}$$ $$\begin{aligned}
&&x=\psi^T\lambda^{-1}\psi,\quad y 
   =\tau^{-1}\mu^2,\quad z=y-\tau^{-1}\psi^T\hat J\tilde\omega \,,\end{aligned}$$ and the $2\times 2$ matrix, $$\begin{aligned}
\hat J= \left(
  \begin{array}{ccccccc}
   0&1\\
   -1&0\\
  \end{array}
 \right) \,.  \end{aligned}$$ We note that this $7\times7$ matrix $M$ is symmetric, $M^T=M$, and unimodular, 