Fibonacci identities and Fibonacci pairs

A Fibonacci pair $F_s(w,x)$ of rank $s$ is a pair $s\times s$ nonsingular matrices such that $wx=xw$ and that the entries of $a{w}^n$ and $ax{w}^m$ are polynomials of Fibonacci or Lucas numbers for some nonzero $a$. We construct identities systematically by the study of $F_2(w,x)$ and $F_3(w,x)$.