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We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. We denote by$\Sigma^{13}_{\mathbb{A}}$ the affine variety obtained from$\Sigma^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we referthe precise definition to Definition 1.1 of the paper). We show that severalweighted projectivizations of $\Sigma^{13}_{\mathbb{A}}$ and$\Sigma^{14}_{\mathbb{A}}$ produce, as weighted complete intersections,examples of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to$24$ classes of the graded ring database. Except No.360 in the database, theseprime $\mathbb{Q}$-Fano threefolds have a Type I Tom projection. Moreover, theyare not weighted complete intersections of the cluster variety of type $C_2$introduced by Coughlan and Ducat. We also show that a partial projectivizationof $\Sigma^{14}_{\mathbb{A}}$ has a $\mathbb{P}^2\times \mathbb{P}^2$-fibrationover the affine space $\mathbb{A}^9$.

Key Varieties for Prime $\mathbb{Q}$-Fano Threefolds related with $\mathbb{P}^2\times \mathbb{P}^2$-Fibrations. Part I