Is this tensor representation of the angular momentum operator correct?

If the position operator is $mathbf R = (tilde X,tilde Y,tilde Z)$ and the momentum operator is $mathbf P = (tilde P_x,tilde P_y,tilde P_z)$ with components
$$
tilde X = Xotimesmathbb I otimesmathbb I,qquad tilde Y = mathbb I otimes Y otimesmathbb I,qquad tilde Z = mathbb I otimes mathbb I otimes Z
tilde P_x = P_xotimesmathbb I otimesmathbb I,qquad tilde P_y = mathbb I otimes P_y otimesmathbb I,qquad tilde P_z = mathbb I otimes mathbb I otimes P_z
$$
These operators (with a twidle) are called the extended from the untwidled operators (as in Tannoudji) and they act on the tensor product of the Hilbert spaces $scr E_x, scr E_y$ and $scr E_z$, i.e., on $scr E = scr E_x otimes scr E_y otimes scr E_z$

then does that the angular momentum operator $mathbf L = (L_x,L_y,L_z)$ have components like
$$
L_x = tilde Ytilde P_z – tilde Ztilde P_y
= mathbb I otimes Y otimes P_z – mathbb I otimes P_y otimes Z
L_y = tilde Ztilde P_x – tilde X tilde P_z
= P_x otimes mathbb I otimes Z – X otimes mathbb I otimes P_z
L_Z = tilde Xtilde P_y – tilde Ytilde P_x
= X otimes P_y otimes mathbb I – P_x otimes Y otimes mathbb I
$$