A Cayley–Hamilton trace identity for 2×2 matrices over Lie-solvable rings

First we construct an algebra satisfying the polynomial identity [[x,y],[u,v]]=0, but none of the stronger identities [x,y][u,v]=0 and [[x,y],z]=0. Then we exhibit a Cayley–Hamilton trace identity for 2×2 matrices with entries in a ring R satisfying [[x,y],[x,z]]=0 and .