  Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a
tracially symmetric quantum Markov semigroup on a C*-algebra $\mathcal{A}$
there exists a densely defined derivation $\delta$ from $\mathcal{A}$ to a
Hilbert bimodule $H$ such that $L^{(2)}=\delta^*\circ \overline{\delta}$. Here
we show that this construction of a derivation can in general not be
generalised to quantum Markov semigroups that are symmetric with respect to a
non-tracial state. In particular we show that all derivations to Hilbert
bimodules can be assumed to have a concrete form, and then we use this form to
show that in the finite-dimensional case the existence of such a derivation is
equivalent to the existence of a positive matrix solution of a system of linear
equations. We solve this system of linear equations for concrete examples using
Mathematica to complete the proof.
