G\in GL_4 (\Z[i]) \hspace{2mm} | \hspace{2mm} G^\dg \,  \begin{pmatrix}
0 & {\bf 1}_2 \\
-{\bf 1}_2 & 0 \\
\end{pmatrix} \, G = \begin{pmatrix}
0 & {\bf 1}_2 \\
-{\bf 1}_2 & 0 \\
\end{pmatrix} \Big\}.$$ ${\bf 1}_2$ denotes the 2 $\times$ 2 identity matrix. $\begin{pmatrix}
A & B \\
C & D \\
\end{pmatrix}$ in the modular group $\Gamma$ acts on $\omega \in {\bf H}_2$ as $$\begin{pmatrix}
A & B \\
C & D \\
\end{pmatrix}\cdot \omega = (A\, \omega+B) (C\, \omega+D)^{-1}.$$ There is an involution, ${\cal T}$, that acts on ${\bf H}_2$ as $${\cal T}\cdot \omega =\omega^t.$$ The group $\Gamma_{\cal T}$ is defined to be the semi-direct product of the modular group $\Gamma$ and $<{\cal T}>$: $$\Gamma_{\cal T} := \Gamma \, \rtimes <{\cal T}>.$$ There is an isomorphism $\Gamma_{\cal T} \cong O^+(L^{2,4})$, and this induces the isomorphism (\[H2 isom in 2.2\]) [@Matsumoto].

Under the isomorphism $\Gamma_{\cal T} \cong O^+(L^{2,4})$, the ring of $O^+(L^{2,4})$-modular forms corresponds to the ring of $\Gamma_{\cal T}$-modular forms of even characteristic [@Vinberg2010], generated by the five modular forms $J_k$ of weights $2k$, $k=2, \cdots, 6$ [@CMSE7]. See [@CMSE7] for definitions of the modular forms $J_k$. In a special situation, the modular forms $J_2, J_3, 