I have always found it a bit difficult to understand the variation of an action written in differential form language. For example, take the action
$$\int tr A\wedge A\wedge A$$
where $A=A_\mu dx^\mu=A^I_\mu T_I dx^\mu$ is a one form valued in some Lie algebra (this is basically one part of a chern simons action). Now the variation is supposed to give the equations of motion $A\wedge A=0$, however, this is far from trivial to me. If I write it out I get
$$\int A^I_\mu A^J_\nu A^K_\rho tr(T_I T_J T_K) dx^\mu\wedge dx^\nu \wedge dx^\rho$$
Now I suppose I want to vary with respect to the $A_\mu^I$ to get something like
$$\int 3(dA^I_\mu) A^J_\nu A^K_\rho tr(T_I T_J T_K) dx^\mu\wedge dx^\nu \wedge dx^\rho$$
It is still far from trivial to me what this means. We can try to write it as a normal integral by using $dx^\mu\wedge dx^\nu \wedge dx^\rho\sim \epsilon^{\mu\nu\rho} dx^3$ to obtain the following equations of motion
$$\epsilon^{\mu\nu\rho}A^J_\nu A^K_\rho tr(T_I T_J T_K)=0\Rightarrow A_\nu^J A_\mu^K tr (T_I T_L c_{JK}^L)=0$$
where the $c$ are the structure constants. Now the trace should define a non-degenerate bilinear form $tr(T_IT_J)=g_{IJ}$, and from this one can indeed find that $A_\nu^J A_\mu^K c_{JK}^L=0$ which indeed is equivalent to $A\wedge A=0$. (I think this calculation is correct but corrections are welcome)
However I find this derivation very cumbersome and complicated to derive such an "obvious" result. So how should I think of variations of actions written with differential forms? what are the rules for doing such variations?
