Strictly paired ${\rm CR}$ linear automorphisms of $\R^4$

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 $\mathrm{GL}(2,\mathbb{C})^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.