Unipotent diagonalization of matrices

An element $u$ of a ring $R$ is called \textsl{unipotent} if $u-1$ is
 nilpotent. Two elements $a,b\in R$ are \textsl{unipotent equivalent}
 there exist unipotents $p,q\in such that $b=q^{-1}ap$. square
 matrices $A,B$ \textsl{strongly unipotent equivalent} there
 triangular $P,Q$ with $B=Q^{-1}AP$.
 In this paper, over commutative reduced rings, we characterize the matrices
 which strongly equivalent to diagonal matrices. For $2\times 2$
 B\'{e}zout domains, nilpotent some multiples $E_{12}$ and nontrivial
 idempotents $E_{11}$.