A How to derive the quantum detailed balance condition?

[lsdragon]

TL;DR Summary

I want some help to get the definition of quantum detailed balance condition from analogy of classical detailed balance condition

In the "On The detailed balance conditions for non-Hamiltonian systems", I learned that for a Markov open quantum system to satisfying the master equation with the Liouvillian superoperators, the detailed balance condition will be

> Definition 2: The open quantum Markovian system ($dim(\mathcal{H}) < \infty$) obeys the detailed balance principle if the generator $L$ in Heisenberg picture is a normal operator in Hilbert space $\mathcal{B}_{\rho_0}(\mathcal{H})$ (see Definition 1).

>Definition 1: $\mathcal{B}_{\rho_0}(\mathcal{H})$ denotes the Hilbert space of all linear operators on the finite-dimensional Hilbert space $\mathcal{H}$ with the scalar product defined by the formula
$$\langle A, B\rangle = Tr(A^\dagger B \rho_0), A,B \in \mathcal{B}_{\rho_0}(\mathcal{H})$$
where $\rho_0$ is a fixed state (density matrix) and $\rho_0 > 0$.

The $L$ is the adjoint operator, defined with respect to definition 1, of the Liouvillian superoperator $\mathcal{L}$, such that
$$
\frac{d \rho}{d t} = \mathcal{L} \rho \\
\frac{d A}{d t} = L A, A\in \mathcal{B}_{\rho_0}(\mathcal{H}).
$$

The author started from the classical detailed balance condition $p_{ij}\pi_j = p_{ji}\pi_i$ and finally get to definition 2.

For me, I will write the quantum analogy of detailed balance as
$$
\langle A,L(B) \rangle = \langle B, L(A)\rangle .
$$
I can not get the normality of $L$ from the above definition.
Then, my question is that how can we get to definition 2 starting from the classical version of detailed balance?