MSG 2.5

\[\begin{align*} &E_{2}^{1,-1}=\mathbb{Z}^3\\ &E_{1}^{0,-1}=0\\ &E_{1}^{1,-1}=2\mathbb{Z}[\boldsymbol{b}^{(1)}_{a_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{b_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{c_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{d_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{e_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{g_1}]\\ &E_{1}^{2,-1}=2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\alpha _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\beta _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\gamma _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\delta _1}] \end{align*}\]

\[\begin{align*} &[X^{(1)}]^{-1}=\left( \begin{array}{ccccccc} a_1 & b_1 & c_1 & d_1 & e_1 & f_1 & g_1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &[V^{(1)}]^{-1}=\left( \begin{array}{ccccccc} a_1 & b_1 & c_1 & d_1 & e_1 & f_1 & g_1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &\Sigma^{(1)}=\left( \begin{array}{ccccccc} 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} \]