Suppose $N = 4$. Given $g \in L^{1}(S^{N-1})$, I would like to know if is it possible to write $$ \int_{S^{N-1}} g(x,y,z,w) d\sigma^{N-1} = C \int_{S^{N-3}}\left( \int_{S^{N-3}} g(x,y, z,w) d \sigma^{N-3}_{(x,y)} \right) \sigma^{N-3}_{(z,w)} \qquad (1) $$ for a suitable constant $C$.
What I think can be used, but was unsure about:
In the book "Fourier analysis and Hausdorff dimension" by Mattila and Pertti (Cambridge University Press, 2015), on page 33, there is the following construction.
Consider $e_1 = (1,0,0,0) \in S^{N-1} \subset \mathbb{R}^4$. For $\theta \in [0, \pi]$ define $$ S_\theta = \left\{ v = (x,y,z,w) \in S^{N-1} : e_1 \cdot v = \cos(\theta) \right\}. $$ The above set seems like it is a sphere in $\mathbb{R}^3$ centered at $C = (\cos\theta) e_1$ with radius $\sin\theta$. This works because the projection of any vector $v$ of $S_\theta$ along the vector $e_1$ is of the form $k e_1$, where $k = x \cdot e_1 = \cos\theta$.
On this sphere, one can use the natural surface measure, which we denote by $\sigma_{\sin(\theta)}^{N-2}$. Mattila says that $\sigma_{\sin(\theta)}^{N-2}(S_\theta) = b (\sin(\theta))^{N-2}$, where $b = \sigma^{N - 2}(S^{N-2}).$ Using this, he proves that for any $g \in L^1(S^{N-1})$ $$ \int_{S^{N-1}} g(v) d \sigma^{N-1}_v = \int_{0}^{\pi} \left( \int_{S_\theta} g(v) d \sigma^{N-2}_{\sin(\theta) v} \right) d \theta. $$
I don't really know if some version of this technique must be used to prove $(1)$. If anyone know another approach, or any hint or reference with things like this, I would be very grateful.