Is there a meaningful way to define the covariant derivative of the connection coefficients, $\Gamma^a_{bc}$? As in, does it make sense to define the object $\nabla_d\Gamma^a_{bc}$? Since the connection coefficients symbol doesn't transform as a tensor, it would seem like there should be some obstruction to defining this in the usual way, treating $a$ as a contravariant index and $b$ and $c$ a covariant indices.
Part of my motivation for thinking about this was for writing the Riemann tensor in terms of this symbol $\nabla_d\Gamma^a_{bc}$. If you work in a local Lorentz frame at a point where $\Gamma^a_{bc}$ all vanish, the expression for the Riemann tensor is just $$R^a_{\phantom{a}bcd}=\partial_c\Gamma^a_{bd}-\partial_b\Gamma^a_{cd}.$$ So then I'd like to "covariantize" this expression for a general coordinate system by writing \begin{equation} R^a_{\phantom{a}bcd}=\nabla_c\Gamma^a_{bd}-\nabla_b\Gamma^a_{cd}. \tag{*} \end{equation} If I pretend that $\Gamma^a_{bd}$ should have a covariant derivative defined by treating the indices as normal tensor indices, I get for this expression something pretty close to the right answer $$R^a_{\phantom{a}bcd}=\partial_c\Gamma^a_{bd}-\partial_b\Gamma^a_{cd} +2(\Gamma^a_{ce}\Gamma^e_{bd}-\Gamma^a_{eb}\Gamma^e_{cd})$$ and curiously, if I define $$\nabla_c\Gamma^a_{bd} \equiv \partial_c\Gamma^a_{bd} + \Gamma^a_{ce}\Gamma^e_{bd}-\Gamma^e_{cd}\Gamma^a_{eb} + \Gamma^e_{cb}\Gamma^a_{ed} $$ where the last term appears with the wrong sign from what you get with an ordinary $(1,2)$ tensor, the expression $(*)$ above for the Riemann tensor is correct. Is this just a coincidence, or is there some reason to define a covariant derivative of the connection symbol like that?
Update: The expression that gives the right form of the Riemann tensor for $(*)$ is actually $$\nabla_c\Gamma^a_{bd} \equiv \partial_c\Gamma^a_{bd} + \Gamma^a_{ce}\Gamma^e_{bd}-\Gamma^e_{cd}\Gamma^a_{eb}$$ so it is as if we are not treating $b$ as a tensor index, and we are just writing the covariant derivative of a $(1,1)$ tensor.