Matrix/Special orthogonal group/Field/Definition

Let ${\displaystyle {}K}$ be a field, and ${\displaystyle {}n\in \mathbb {N} _{+}}$. An orthogonal ${\displaystyle {}n\times n}$-matrix ${\displaystyle {}M}$ fulfilling

${\displaystyle {}\det M=1\,}$

is called a special orthogonal matrix. The set of all special orthogonal matrices is called special orthogonal group; it is denoted by ${\displaystyle {}\operatorname {SO} _{n}\!{\left(K\right)}}$.