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A direct determinant computation givesThe zero eigenvalue has algebraic multiplicity two but a one-dimensional eigenspace, so it gives one Jordan block of size two. The other block is $J_1(4)$. For an explicit Jordan basis, takeThese are linearly independent and satisfy $Dv=0$, $Du=v$, $Dw=4w$. ThusThe first summand has the chain $u\mapsto v\mapsto 0$ and is an indecomposable module; the other is simple.

If $K^2\ne W$, the three eigenvalues are distinct, so a basis of eigenvectors givesThe transformed coordinates then have the displayed exponential form. If $K^2<W$, this is a complex eigenbasis; real concentrations are recovered by conjugate-paired coefficients. A real basis instead gives a decaying two-dimensional rotation block with decay rate $-K$ and angular frequency $\sqrt{W-K^2}$.

The proposed diagonalization is not valid for every positive choice of rates. For example $ϵ=\delta =1$, $\eta =4$ gives $K=-3$, $W=9$, with repeated eigenvalue $-3$ and only one corresponding eigenvector. More generally the defective repeated eigenvalue in a cyclic conversion model occurs when $K^2=W$. A Jordan basis then givesThe extra factor $t$ cannot be represented by two independent pure exponential coordinates in a diagonal basis.