Because those derivatives aren't acting on T, they are acting on [itex]\omega_c[/itex], and hence [itex]\frac{1}{6}\nabla_{a} \nabla_b \omega_{c}[/itex] will just be some tensor.
In contrast, if you look at the term [itex]\frac{1}{6}\nabla_{a} \omega_c \nabla_b [/itex], only the first operator acts on [itex]\omega_c[/itex]; [itex]\nabla_{b}[/itex] will act on whatever tensor comes to the right of this expression.
