  We show that the generator of a GNS-symmetric quantum Markov semigroup can be
written as the square of a derivation. This generalizes a result of Cipriani
and Sauvageot for tracially symmetric semigroups. Compared to the tracially
symmetric case, the derivations in the general case satisfy a twisted product
rule, reflecting the non-triviality of their modular group. This twist is
captured by the new concept of Tomita bimodules we introduce. If the quantum
Markov semigroup satisfies a certain additional regularity condition, the
associated Tomita bimodule can be realized inside the $L^2$ space of a bigger
von Neumann algebra, whose construction is an operator-valued version of free
Araki-Woods factors.
