Torsion-free, symmetric connection and non-coordinate basis

Question

The torsion tensor is defined as (Hawking p.34) \begin{equation} \mathbf{T}(\mathbf{X},\mathbf{Y}) = \nabla_{\mathbf{X}}\mathbf{Y} - \nabla_{\mathbf{Y}}\mathbf{X} - [\mathbf{X},\mathbf{Y}]. \end{equation} The connection is defined as (Hawking p.31) \begin{equation} \Gamma^a{}_{bc} = \langle\mathbf{E}^a,\nabla_{\mathbf{E}_b}\mathbf{E}_c\rangle, \end{equation} where $\{\mathbf{E}_a\}$ is any vector basis. So we have \begin{equation} T^c{}_{ab} = \langle \mathbf{E}^c,\mathbf{T}(\mathbf{E}^a,\mathbf{E}^b) \rangle = \Gamma^c{}_{ab}-\Gamma^c{}_{ba} - \langle \mathbf{E}^c,[\mathbf{E}^a,\mathbf{E}^b] \rangle \end{equation} In coordinate basis the Lie bracket (commutator) vanishes, but in general the commutator coefficients do not vanish, e.g. in the non-coordinate basis.

Let $\mathbf{T}=0$. Does it mean that the connection is symmetric only in the coordinate basis?

On the other hand, we can calculate The torsion tensor in holonomic coordinate as \begin{equation} T^\sigma{}_{\mu\nu} = \Gamma^{\sigma}{}_{\mu\nu} - \Gamma^{\sigma}{}_{\nu\mu}, \end{equation} or in an orthonormal frame (indicating with Latin indices) \begin{equation} T^c{}_{ab} = \Gamma^{c}{}_{ab} - \Gamma^{c}{}_{ba}-e^\mu_ae^\nu_b(e^c_{\mu,\nu}-e^c_{\nu,\mu}). \end{equation} Can we show they are equivalent under change of basis? The problem is usually we write tensor equation with abstract index notation, and I just found that they give different results with different a priori chosen basis. Does it mean that we have to express the tensor equation with non-coordinate basis, since it seems to be more general?

EDIT

This should be a fairy straightforward tensor manipulation (I should have put more effort).

Define $\partial_a \mathbf{e}_b = \Gamma^c{}_{ab}\mathbf{e}_c$ where the Latin indices denote orthonormal frame. Then \begin{equation} \partial_a \mathbf{e}_b = e^\mu_a \nabla_\mu (e^\nu_b e_\nu) = [e^\nu_a(\partial_\mu e^\sigma_b) + e^\mu_a e^\nu_b \Gamma^\sigma{}_{\mu\nu}]e^c_\sigma e_c, \end{equation} that is \begin{equation} \Gamma^c{}_{ab} = e^\mu_a e^c_\sigma \partial_\mu e^\sigma_b + e^\mu_a e^\nu_b e^c_\sigma \Gamma^\sigma{}_{\mu\nu}. \end{equation} Plug this into $e^\mu_a e^\nu_b e^c_\sigma T^\sigma{}_{\mu\nu}$ we can show it is equal to $T^c{}_{ab}$.

The question remains: to manipulate these tensor object we use abstract index notation. Given a tensor equation like $\mathbf{T}(\mathbf{X},\mathbf{Y}) = \nabla_{\mathbf{X}}\mathbf{Y} - \nabla_{\mathbf{Y}}\mathbf{X} - [\mathbf{X},\mathbf{Y}]$, how do we write it in abstract index notation? The notation of symmetric relies on the index placement of the component. For $\mathbf{T}=0$ it give symmetric $\Gamma^\sigma{}_{\mu\nu}$ but not to the $\Gamma^a{}_{bc}$. Are they all equal for whatever basis we choose to express, and differ only in physical expression (like what observer sees)? However, on page 24 of the book by Wald, the author states (relating to abstract index notation)

However, in some cases it will be convenient to use a particular type of basis, e.g., a coordinate basis adapted to the symmetries of a particular spacetime. If we do this, then the equations we write down for the tensor components may be valid only in this basis.

Answer 1

Non-coordinate bases are always used when coupling to fermions. You could simply look at the maths literature, but as a physicist you'll want to look at textbooks on supergravity. Take a look at words such as "vielbein", "frame" and "spin connection". As to literature, there's Freedman/van Proeyen "Supergravity", chapter 7. Alternatively Ortin's "Gravity and Strings". Also note that torsion shows up in supergravity due to the presence of the Fermions. So that's also nicely discussed in those books.

I would have posted this as a comment, yet apparently my reputation score here is insufficient for doing so.