QM- Proving the uncertainty relation - Commutators

Folks,

I am stuck on the derivation of the uncertainty relation when using the commutator and anti commutator...

given$\displaystyle \triangle A=\hat A -\langle A \rangle $

$\displaystyle

\displaystyle \triangle A \triangle B=\frac{1}{2}[\triangle A, \triangle B]+\frac{1}{2}\left(\triangle A, \triangle B\right)

$

Above on the RHS is the commutator and anti-commutator respectively. I dont understand the next line for the commutator

$\displaystyle

[\triangle A, \triangle B]_\pm=[\hat A -\langle A \rangle , \hat B -\langle B \rangle ]_\pm=[\hat A, \hat B -\langle B \rangle ]_\pm-[\langle A \rangle , \hat B-\langle B \rangle ]_\pm$

the last term in the above line is 0 because <A> is a c number

$\displaystyle

=[\hat A, \hat B]-[\hat A, \langle B \rangle ]_\pm

$

where the last term in the above line is also 0 for same reason.

Now I do know [A,B] is AB-BA but I dont know where this algebaic derivation is coming out of..

Thanks