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For the given monomial $Y=\frac{x_{i_1}\cdots x_{i_k}}{x_{j_1}\cdots x_{j_k}}$ the coefficient $L(Y)$ multiplied by the constant $(-1)^{\sum_{i<j} a_{ij}}$ equals $$[Y]\prod_{i,j}(1-x_i/x_j)^{a_{ij}}=\int Y^{-1}d\mu,$$ where $d\mu$ is the measure on the $4$-dimensional torus $\mathbb{T}^4=\{(x_1,x_2,x_3,x_4)\in \mathbb{C}^4:|x_1|=|x_2|=|x_3|=|x_4|=1\}$ which density w.r.t. normalized Lebesgue measure equals to $\prod_{i,j}(1-x_i/x_j)^{a_{ij}}$ (the key observation is that this is always real and almost always positive). Your matrix is then Gram matrix of the functions $1,x_2/x_1,x_4/x_3,x_2x_4/x_1x_3$ in $L^2(\mu)$. They are linearly independent, hence the determinant is strictly positive.