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 $S O (3)$ , is the set of all $3 \times 3$ matrices $𝑀 \in ℝ 3 \times 3$ satisfying

$𝑀 𝖳 𝑀 = 𝐈$
$d e t 𝑀 = 1$

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

Lie algebra

Following Keppeler's Lie algebra convention, the Lie algebra $𝔰 𝔬 (3)$ of $S O (3)$ is given by imaginary hermitian matrices with the bracket $- 𝑖 [-, -]$ , where $[-, -]$ denotes the matrix commutator. A suitable basis is

𝐽 1 = ⎡ ⎢ ⎢ ⎣ 000 00 - 𝑖 0 𝑖 0 ⎤ ⎥ ⎥ ⎦, 𝐽 2 = ⎡ ⎢ ⎢ ⎣ 00 𝑖 000 - 𝑖 00 ⎤ ⎥ ⎥ ⎦, 𝐽 3 = ⎡ ⎢ ⎢ ⎣ 0 - 𝑖 0 𝑖 00 000 ⎤ ⎥ ⎥ ⎦

which gives the Structure constants

- 𝑖 [𝐽 𝑗, 𝐽 𝑘] = 3 \sum ℓ = 1 𝜖 𝑗 𝑘 ℓ 𝐽 ℓ

where $𝜖$ is the Levi-Civita symbol.

Properties

Exponentiation gives the Axis-angle parameterization $𝑒 - 𝑖 𝛼 ⃗ 𝑛 \cdot ⃗ 𝐽 = R ⃗ 𝑛 (𝛼)$
Irreps of $S O (3)$

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