Orthonormal Families, Standard Basis Vectors, and Plane Rotations of Euclidean Space

Let $n$ be a natural number and let $\mathbb{R}^{n}$ be Euclidean space with the dot product $v\cdot u$.

Orthonormal family. Vectors $w_1,\dots,w_p\in\mathbb{R}^{n}$ (with $p$ a natural number) form an orthonormal family if

Standard basis vectors. For $1\le i\le n$, the standard basis vector $e_i\in\mathbb{R}^{n}$ is the vector whose $i$-th coordinate is $1$ and whose other coordinates are $0$. The family $e_1,\dots,e_n$ is orthonormal, and for every $v\in\mathbb{R}^{n}$ the $i$-th coordinate of $v$ equals $v\cdot e_i$.

Plane rotations. Let $n\ge2$, let $1\le i<j\le n$, and let $a,b$ be real numbers with $a^{2}+b^{2}=1$. The plane rotation of $\mathbb{R}^{n}$ in coordinates $(i,j)$ with parameters $(a,b)$ is the map $g:\mathbb{R}^{n}\to\mathbb{R}^{n}$ defined coordinatewise by

Compositions. A finite composition of plane rotations of $\mathbb{R}^{n}$ is a map of the form $h=g_L\circ\dots\circ g_1$ where $L$ is $0$ or a natural number and each $g_l$ is a plane rotation of $\mathbb{R}^{n}$; for $L=0$ the empty composition is defined to be the identity map of $\mathbb{R}^{n}$.