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 ${}_b\partial_a+{}_a\partial_b = 0$.