nonlinear transform of Gaussian random variable that preserves Gaussianity

I recently know that following results.

suppose that $x_1, x_2, x_3$ are independent real Gaussian random variables with $\mathcal{N}(0, 1)$. Then

$$
\frac{x_1 + x_2 x_3}{\sqrt{1+x_3^2}} \sim \mathcal{N}(0, 1)
$$

We can prove this result by direct computing. But I am wondering if there is a simpler way. Also, since this result is interesting. I am wondering if there is any generalization

Thanks

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The point is that the conditional distribution of your random variable given $x_3$ is always ${\cal N}(0,1)$. One generalization is this. Suppose
$X_1, \ldots, X_n$ are independent ${\cal N}(0,1)$ random variables, and ${\bf Y} = (Y_1, \ldots, Y_n)$ is a vector-valued random variable independent of $X_1, \ldots, X_n$ and supported on the sphere $Y_1^2 + \ldots + Y_n^2 = 1$. Then
${\bf X} \cdot {\bf Y} = X_1 Y_1 + \ldots + X_n Y_n \sim {\cal N}(0,1)$.