Consider three independence random variables $X,Y,Z$, where $X,Z$ are each distributed as the first coordinate of a unit circle in $\mathbb{R}^2$ and $Y$ is distributed as the first coordinate of a unit $S^2$ in $\mathbb{R}^3$. Then it is true that $XZY + XZ$ is distributed as the first coordinate in $S^2$ again!

I know this is true by looking at the matrix product $\gamma_{1,2}(\theta_1) \gamma_{2,3}(\theta_2) \gamma_{1,3}(\theta_3)$ where $\gamma_{i,j}(\theta)$ is the rotation matrix by $\theta$ in the $i \wedge j$ plane in $\mathbb{R}^3$. But is there a probabilistic way to prove it using beta-gamma algebra or otherwise?