I have a question about Kac's book.
On page 331, 14.2 and 14.3, let $A$ be an affine type matrix such that ${}^tA = X_\ell^{}(1)$ and let $g$ be a simple finite dimensional Lie algebra of type $X_{\ell}$. Let $j \in \mathbb{Z}_+$, $c_j$ the number of roots of $g$ of height $j$, $M_j$ the number of times $j$ appears in $E_0$. Then \begin{align} & c_j + c_{j-1} - M_{j-1}, M_{j} = M_{h-j}, (1) \\ & c_j + c_{h-j} = \ell + M_j, (2) \\ & c_j + c_{h+1-j} = \ell, (3) \\ & c_j + c_{h+2-j} = \ell - M_{j-1}. (4) \end{align}
Let $L(\Lambda)$ be an integrable $g(A)$-module of level 1. It is said that the following followings from (14.4.4) and (1)--(4). \begin{align} & \dim_q L(\Lambda) = \prod_{j \geq E_0} \prod_{n \in \mathbb{Z}_+} (1-q^{j+nh})^{-1}, (i) \\ & \dim_q L(2\Lambda) = \prod_{j \geq E_0} \prod_{n \in \mathbb{Z}_+} (1-q^{j+nh})^{-1}(1-q^{j+1+n(h+2)})^{-1}. (ii) \end{align}
The formula (14.4.4) is \begin{align} \dim_q L(\Lambda_0) = \prod_{j \geq 1} (1-q^j)^{-\dim s_j}, \end{align} where $s = \oplus_{j \in \mathbb{Z}} s_j$ is the principal subalgebra of $g'(A)$.
How to prove (i) and (ii)? Any hint would be greatly appreciate!