Calculate curvature by coordinate component method

[PhyPsy]

I'm trying to follow the math in Wald's General Relativity where he starts out with the equation for covariant derivative:
$\nabla$_b$\omega$_c = $\partial$_b$\omega$_c - $\Gamma$^d_bc$\omega$_d

He uses that to derive the equation for a double covariant derivative:
$\nabla$_a$\nabla$_b$\omega$_c = $\partial$_a($\partial$_b$\omega$_c - $\Gamma$^d_bc$\omega$_d) - $\Gamma$^e_ab($\partial$_e$\omega$_c - $\Gamma$^d_ec$\omega$_d) - $\Gamma$^e_ac($\partial$_b$\omega$_e - $\Gamma$^d_be$\omega$_d)

Now, using the Riemann tensor definition R_abc^d$\omega$_d = $\nabla$_a$\nabla$_b$\omega$_c - $\nabla$_b$\nabla$_a$\omega$_c, this equation is derived:
R_abc^d$\omega$_d = $\partial$_[a$\partial$_b]$\omega$_c - $\omega$_d$\partial$_[a$\Gamma$^d_b]c - $\Gamma$^d_c[b$\partial$_a]$\omega$_d - $\Gamma$^e_[ab]($\partial$_e$\omega$_c - $\Gamma$^d_ec$\omega$_d) - $\Gamma$^e_c[a$\partial$_b]$\omega$_e - $\Gamma$^e_c[a$\Gamma$^d_b]e$\omega$_d

I know that the term $\partial$_[a$\partial$_b]$\omega$_c cancels, and, due to the symmetry of the metric connection, $\Gamma$^e_[ab]($\partial$_e$\omega$_c - $\Gamma$^d_ec$\omega$_d) also cancels, but the next step in the book also has a couple other terms canceled out:
$\Gamma$^d_c[b$\partial$_a]$\omega$_d, and
$\Gamma$^e_c[a$\partial$_b]$\omega$_e

I don't see how these terms cancel. Can someone help me?

 

[fresh_42]

We can substitute $e$ by $d$ without changing anything. Now what is left is the condition ${}_b\partial_a+{}_a\partial_b = 0$.