Proof of Plane Rotations Preserve the Dot Product

Let $g$ be the plane rotation in coordinates $(i,j)$ with parameters $(a,b)$, $a^{2}+b^{2}=1$. By the definition of the dot product, $g(v)\cdot g(u)$ is the sum over coordinates of the products of corresponding coordinates. The coordinates with index $k\notin\{i,j\}$ contribute $v_ku_k$, unchanged. The coordinates $i$ and $j$ contribute

Summing all contributions gives $g(v)\cdot g(u)=v\cdot u$.

For a finite composition $h=g_L\circ\dots\circ g_1$, the identity $h(v)\cdot h(u)=v\cdot u$ follows by applying the single-rotation identity $L$ times (and holds trivially for the empty composition, which is the identity map). Finally, if $w_1,\dots,w_p$ is an orthonormal family, then $h(w_i)\cdot h(w_i)=w_i\cdot w_i=1$ and $h(w_i)\cdot h(w_k)=w_i\cdot w_k=0$ for $i<k$, so $h(w_1),\dots,h(w_p)$ is an orthonormal family. $\blacksquare$