In the WZW model, you have a 2 dimensional manifold $\Sigma$ and it's 3 dimensional extension $\tilde{\Sigma}$. In the WZW action, there is a term of the form: $S_{1}[g] = \alpha\int_{\tilde{\Sigma}} Tr[g^{-1}dg \wedge g^{-1}dg \wedge g^{-1}dg]$, where $g: \tilde{\Sigma} \rightarrow SU(2)$.
I tried to vary this with respect to $g$ and I got this using the product rule, which I hope is correct: $\delta S_1 [g] = \alpha\int_{\tilde\Sigma} Tr[\delta(g^{-1}dg)\wedge g^{-1}dg\wedge g^{-1}dg + g^{-1}dg\wedge \delta(g^{-1}dg)\wedge g^{-1}dg + g^{-1}dg\wedge g^{-1}dg\wedge \delta(g^{-1}dg)]$.
My question is how do I treat the expression in the brackets of the trace? Can I apply the cyclic property of the trace freely with forms in the brackets? can I swap the components and use the anti-symmetry of the wedge product freely?