MSG 88.82

\[\begin{align*} &E_{2}^{1,-1}=0\\ &E_{1}^{0,-1}=2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{B}_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{C}_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_3}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_4}]\\ &E_{1}^{1,-1}=2\mathbb{Z}[\boldsymbol{b}^{(1)}_{b_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{b_2}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{c_1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_1}]\\ &E_{1}^{2,-1}=2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\alpha _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\beta _1}]\oplus2\mathbb{Z}[\boldsymbol{b}^{(2)}_{\gamma _1}] \end{align*}\]

\[\begin{align*} &[X^{(1)}]^{-1}=\left( \begin{array}{cccc} b_1 & b_2 & c_1 & f_1 \\ 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ \end{array} \right)\\ &[V^{(0)}]^{-1}=\left( \begin{array}{ccccc} \text{B}_1 & \text{C}_1 & \text{D}_1 & \text{D}_3 & \text{D}_4 \\ 1 & 0 & 0 & -1 & -1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &\Lambda^{(0)}=\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array} \right)\\ &[V^{(1)}]^{-1}=\left( \begin{array}{cccc} b_1 & b_2 & c_1 & f_1 \\ 1 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right)\\ &\Sigma^{(1)}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} \]