Plane Rotations Preserve Independent Standard Normal Families

Let $m$ be a natural number with $m\ge2$, let $Z_1,\dots,Z_m$ be independent standard normal random variables on a probability space $(\Omega,\mathcal{F},P)$, let $1\le i<j\le m$, and let $a,b$ be real numbers with $a^{2}+b^{2}=1$. Define

equivalently, $\bigl(W_1,\dots,W_m\bigr)(\omega)=g\bigl((Z_1,\dots,Z_m)(\omega)\bigr)$ for every $\omega\in\Omega$, where $g$ is the plane rotation of $\mathbb{R}^{m}$ in coordinates $(i,j)$ with parameters $(a,b)$.

Then $W_1,\dots,W_m$ are independent standard normal random variables on $(\Omega,\mathcal{F},P)$.