MSG 83.48

\[\begin{align*} &E_{2}^{1,-1}=0\\ &E_{1}^{0,-1}=\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{E}_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{E}_2}]\\ &E_{1}^{1,-1}=2\mathbb{Z}[\boldsymbol{b}^{(1)}_{c_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{c_2}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{d_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{d_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_1}]\\ &E_{1}^{2,-1}=2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\beta _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\beta _2}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\gamma _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\delta _1}] \end{align*}\]

\[\begin{align*} &[X^{(1)}]^{-1}=\left( \begin{array}{ccccc} c_1 & c_2 & d_1 & d_2 & f_1 \\ 1 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & -1 & 1 & 0 & 1 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 \\ \end{array} \right)\\ &[V^{(0)}]^{-1}=\left( \begin{array}{cc} \text{E}_1 & \text{E}_2 \\ 1 & 0 \\ 0 & 1 \\ \end{array} \right)\\ &\Lambda^{(0)}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)\\ &[V^{(1)}]^{-1}=\left( \begin{array}{ccccc} c_1 & c_2 & d_1 & d_2 & f_1 \\ 1 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & -1 & 1 & 0 & 1 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &\Sigma^{(1)}=\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} \]