Following the conventions of "Quantum Field Theory and the Standard Model" by Schwartz, we have that for Yang-Mills Theory, an infinitesimal gauge transformation acts like
$$\delta_{\alpha} A = d\alpha - i\left[A, \alpha\right].$$
I am trying to compute the commutator of two gauge transformations, which I expect to give
$$\left[\delta_{\alpha},\delta_{\beta}\right]A = i\delta_{\left[\alpha, \beta\right]}A.$$
However, this is not what I find. Carrying out the computation, I find that
$\left[\delta_{\alpha},\delta_{\beta}\right]A = \delta_{\alpha}\delta_{\beta}A - \delta_{\beta}\delta{_\alpha}A = \delta_{\alpha}\left(d\beta - i\left[A, \beta\right]\right) - \delta_{\beta}\left(d\alpha - i\left[A, \alpha\right]\right) = d\alpha - i\left[d\beta - i\left[A, \beta\right], \alpha \right] - d\beta + i\left[d\alpha - i\left[A, \alpha\right],\beta\right] = d\alpha -d\beta - i\left[ d\beta, \alpha\right] - \left[\left[A,\beta\right],\alpha\right] + i\left[d\alpha, \beta\right] + \left[ \left[A,\alpha\right], \beta \right]$
Some of the commutators can be simplified by realizing that
$\left[d\alpha, \beta\right] - \left[d\beta, \alpha\right] = d\alpha\beta - \beta d\alpha - d\beta \alpha + \alpha d\beta = d\left(\alpha\beta\right) - d\left(\beta\alpha\right) = d\left[\alpha, \beta\right]$.
We can also use the Jacobi identity to see that
$\left[\left[A,\alpha\right], \beta\right] - \left[\left[A, \beta\right], \alpha\right] = \left[\left[A,\alpha\right], \beta\right] + \left[\left[\beta, A\right], \alpha\right] = -\left[\left[\alpha, \beta\right], A\right]$.
Putting everything together, we have that
$\left[\delta_{\alpha},\delta_{\beta}\right]A = d\alpha - d\beta + id\left[\alpha, \beta\right] + \left[A, \left[\alpha, \beta\right]\right] = i\delta_{\left[\alpha, \beta\right]}A + d\alpha - d\beta$.
My question is why has the extra $d\alpha - d\beta$ appeared? Am I carrying out some step in the computation incorrectly, or am I missing something conceptually? As I check, I also computed the commutator by starting with the identity
$e^{i\delta_{\alpha}}e^{i\delta_{\beta}}e^{-i\delta_{\alpha}}e^{-i\delta_{\beta}}A = (1 - \left[\delta_{\alpha}, \delta_{\beta}\right])A + \mathcal{O}(\alpha^2)$.
Here I applied the finite gauge transformations on the left hand side, expanded to second order in $\alpha$ and $\beta$, and matched terms with the right hand side. After doing so I found $\left[\delta_{\alpha},\delta_{\beta}\right]A = i\delta_{\left[\alpha, \beta\right]}A$, as expected, so I'm fairly certain the extra $d\alpha - d\beta$ terms should not be present, but I don't understand where my mistake is when I start from the infinitesimal case.
