(I'm a trying to build a specific gauge connection for a physical theory, but it turns out I need to use a kind of spherical basis for the Lie algebra, and it's confusing me. I'm trained in general relativity but self-taught in gauge theory, and I'm trying to grasp the fine mathematical details. Here is one point of confusion, the solution to which might help me a lot.)
Let $\{\tau_1,\tau_2,\tau_2\}$ be a basis for $\mathfrak{su}(2)$, such that the commutation relations are given by $$[\tau_i,\tau_j]=\varepsilon_{ijk}\tau_k$$ for $\varepsilon_{ijk}$ the anti-symmetric symbol. If we define $$\omega=-\tau_2d\theta+\big(\tau_1\sin\theta+\tau_3\cos\theta\big)d\phi,$$ it is not hard to see that the structure equations are obeyed: In components, if $\omega=A_\mu dx^\mu$, then we get $$\partial_\theta A_\phi-\partial_\phi A_\theta+[A_\theta,A_\phi]=\tau_1\cos\theta-\tau_3\sin\theta+[\tau_1,\tau_2]\sin\theta-[\tau_2,\tau_3]\cos\theta=0.$$ Therefore $\omega$ is a Maurer-Cartan form.
We can use this $\tau$ basis as if it were 'Cartesian', to construct a 'unit spherical' basis for the Lie algebra: Let $$\hat{\tau}_r=\tau_1 \sin\theta\cos\phi+\tau_2 \sin\theta\sin\phi+\tau_3 \cos\theta,$$ $$\hat{\tau}_\theta=\tau_1 \cos\theta\cos\phi+\tau_2 \cos\theta\sin\phi-\tau_3 \sin\theta,$$ $$\hat{\tau}_\phi=-\tau_1 \sin\phi+\tau_2 \cos\phi.$$ (Note that these are implied by $$\bar{\tau}_\alpha=\frac{\partial x^i}{\partial r^\alpha}\tau_i,$$ with $x^i=(x,y,z)$, $\tau_i=(\tau_1,\tau_2,\tau_3)$, $r^\alpha=(r,\theta,\phi)$, and $\bar{\tau}_\alpha=(\tau_r,\tau_\theta,\tau_\phi)=(\hat{\tau}_r,r\hat{\tau}_\theta,r\sin\theta\hat{\tau}_\phi)$.)
It is not hard to show that these obey the same commutation relations as the original basis. Moreover, if we define $U=e^{\phi\tau_3}e^{\theta\tau_2}$, then the gauge rotation $U\tau U^{-1}$ takes $$\tau_1\mapsto\hat{\tau}_\theta,$$ $$\tau_2\mapsto\hat{\tau}_\phi,$$ $$\tau_3\mapsto\hat{\tau}_r;$$ and then with the induced gauge transformation $\omega\mapsto U\omega U^{-1}+UdU^{-1}$, we obtain $$\omega\mapsto-\hat{\tau}_\phi d\theta+\hat{\tau}_\theta \sin\theta d\phi.$$ However, this no longer seems to obey the Maurer-Cartan equations: $$\partial_\theta A_\phi-\partial_\phi A_\theta+[A_\theta,A_\phi]=\partial_\theta\big(\hat{\tau}_\theta \sin\theta\big)+\partial_\phi\hat{\tau}_\phi+[\hat{\tau}_\theta,\hat{\tau}_\phi]\sin\theta=-\tau_r \sin\theta\neq0.$$
I'd have thought that the structure equations would have been invariant under gauge transformations. What am I missing? Thanks in advance for enlightening me.
THOUGHT: Rather than using $\partial_\mu$ here, should I be using some sort of covariant derivative with a term which mimics the Levi-Civita connection, but on the unit sphere on which $\hat{\tau}_\theta$ and $\hat{\tau}_\phi$ live?