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