Strictly paired $\mathrm{C}\mathrm{R}$ linear automorphisms of ${\mathbb{R}}^4$

This paper investigates the algebraic and differential geometric properties of smooth mappings between domains in ${\mathbb{C}}^2$, classifying them according to the constant ranks of their holomorphic and antiholomorphic derivative components. Particular emphasis is placed on strictly paired CR (SPCR) linear automorphisms of ${\mathbb{R}}^4$, where both block matrices $A$ and $B$ have rank 1. We characterise the set of SPCR linear automorphisms as a 12-dimensional submanifold of $\mathrm{G}\mathrm{L}(2,\mathbb{C}{)}^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.