# Problem with expansion of normal ordering

I am reading normal ordering..and far now I'm able to understand. I am stuck in third line from second expression in the book Lectures On Quantum Field Theory By Ashok Das in page no. 237.

It is given that

\begin{aligned} \,&[\;:\!\phi(x)\phi(y)\!:,\phi^{(-)}(z)]\\ =& [\phi(x)\phi(y) + i G^{(+)}(x-y), \phi^{(-)}(z)] \\ = & [\phi^{(+)}(x),\phi^{(-)}(z)]\phi(y) + \phi(x) [\phi^{(+)}(y),\phi^{(-)}(z)] \end{aligned}\tag{6.89}

I understand it is derived as

I understand that Term highlighted in red commute and zero. Is this correct or am I doing anything wrong. Does G term commutes with field as I have highlighted in first.

• The contraction $$\langle 0|\phi(x)\phi(y)|0\rangle=-iG^{(+)}(x-y)$$ commutes with anything, cf. eq. (6.82). For more details, see my related Phys.SE answer here.
• $$\phi = \phi^{(+)} + \phi^{(-)}$$, cf. eq. (6.83).
• The creation parts $$\phi^{(-)}(x)$$ and $$\phi^{(-)}(y)$$ commute.