Remarks on the decomposition of Dirichlet forms on standard forms of von Neumann algebras

For a bounded generator $G$ of a $weakly*$-continuous, completely positive, KMS-symmetric Markovian semigroup on a von Neumann algebra $M$ acting on a separable Hilbert space $H$⁠, let $H$ be the operator induced by $G$ via the symmetric embedding of $M$ into $H$⁠. We decompose the Dirichlet form associated with $H$ into a direct integral of forms whose associated generators are divergences of derivations. Moreover, if the derivations are inner, then the Dirichlet form can be written as the form given by Park [Infinite Dimen. Anal. Quantum Probab., Relat. Top. 3, 1 (2000); 8, 179 (2005)].