A question about reduced density matrices

[naima]

We have $\rho$ and a hamiltonian K on $H_s \otimes H_E$.
have we $$(K \rho)_S \otimes Id_E = K (\rho _S \otimes Id_E)$$ ?

here $\rho _ s$ and $(K \rho) _ s$ are the reduced density matrices.
If P maps an operator O to $O_S \otimes Id_E$, I have to prove that
$PK \rho = KP \rho$ for all $\rho$

 

[naima]

If P maps an operator O to $O_S \otimes Id_E$, I have to prove that
$PK \rho = KP \rho$ for all $\rho$

This link gave mea this idea:
$\partial_t P\rho = i[P\rho, K] = \partial_t (\rho_S \otimes Id_E) = \partial_t \rho_S \otimes Id_E = P\partial_t \rho$ as trace and derivation commute.
$= iP[\rho, K]$
So $PK \rho = KP \rho$
Is it correct?