Pullback invariance of $SO(4)$ transformations for the Haar measure in $SU(2)$

Question

I'm studying the construction of the Haar measure in $SU(2)$ using the $3-sphere \ S^3$ and the tools of the manifolds calculus , geometrically i've the idea of we can think of the group $SO(4)$ as being the set special of orthogonal linear transformations $ L :\mathbb{R^{4}} \rightarrow :\mathbb{R^{4}}$ with $det(L) = 1$.
Now that we have shown each $L \in SO(4)$ is orientation preserving we get that $$ \int_{S^3} L^* (\eta ) = \int_{S^3} \eta $$ for any $3$-form on $S^3$.
The reference is stokes theorem from Spivak's book calculus on manifols, the reference text is Fourier analysis in SU(2) on pages 51-52. I don't understand why this theorem is only enough if when enunciating it, it is integrated over two different sets. The following theorem helps me to understand it more.
Let $X′$ and $X$ be oriented $n$-dimensional manifolds and $f :X' \rightarrow X$ an orientation preserving diffeomorphism. If $W$ is an open subset of $X$ and $W′ = f^{−1}(W)$ $$ \int_{W′} f^∗\omega =\int_{w}ω $$ for all $\omega \in \Lambda ^{n}_c(X)$ (the set of differential $n$-forms).
I have already shown that $L$ it preserves a $S^3$ and that it also preserves the orientation for any basis of $S_x^3$ and the topology induced we see that $S^3$ is open (because is the whole space) and of course $L$ it is a diffeomorphism.
It's what I want to use to make the construction more formal, although I'm not sure if I'm complicating the proof using the last theorem.
If you have any comments that help me understand why Stokes is enough I would appreciate it or if the last theorem is not possible to apply, I would appreciate it.

Answer 1

Use Stokes’s Theorem to prove that if $f$ and $g$ are homotopic maps from $X$ to $Y$ (with $X$ compact, oriented, $k$-dimensional), then for any closed $k$-form $\omega$ on $Y$, we have $$\int_X f^*\omega = \int_X g^*\omega.$$