Real special orthogonal group

Real special orthogonal group of dimension 3

The real special orthogonal group of dimension 3, also called the group of 3D rotations, and typically denoted , is the set of all matrices satisfying

See real special orthogonal group for a discussion of basic properties.

Lie algebra

Following Keppeler's Lie algebra convention, the Lie algebra of is given by imaginary hermitian matrices with the bracket , where denotes the matrix commutator. A suitable basis is

which gives the Structure constants

where is the Levi-Civita symbol.

Properties

1. Exponentiation gives the Axis-angle parameterization
2. Irreps of

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