I have seen in various places the following claim:

Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{X_n}{Z}\right) $$ is a uniform random vector on $S^{n-1}$, where $Z = \sqrt{X_1^2 + \cdots + X_n^2}$.

Many sources claimed this fact follows easily from the orthogonal-invariance of the normal distribution, but somehow I couldn't construct a rigorous proof. (one such "sketch" can be found here).

How to prove this rigorously?

**Edit:**

It has been brought to my attention that this question was already asked before here. However, I find the answer there to be incomplete-it shows that $X$ is orthogonally-invariant, but does not explicitly explains why that implies it is uniform. Therefore I think there is value in keeping this copy as well, as I guess we cannot transfer the answer.

In my question, I explicitly asked for a complete rigorous proof-and I find the answer there to be incomplete.