Orthonormal bases

As a principal homogeneous space

• The set of orthonormal bases for ${\displaystyle \mathbb {R} ^{p,q}}$ is a G-torsor for ${\displaystyle G={\text{O}}(p,q)}$.
• The set of orthonormal bases for ${\displaystyle \mathbb {C} ^{n}}$ is a G-torsor for ${\displaystyle G={\text{U}}(n)}$.
• The set of orthonormal bases for ${\displaystyle \mathbb {C} ^{p,q}}$ is a G-torsor for ${\displaystyle G={\text{U}}(p,q)}$.
• The set of right-handed orthonormal bases for ${\displaystyle \mathbb {R} ^{n}}$ is a G-torsor for ${\displaystyle G={\text{SO}}(n)}$