e(3) — Euclidean motion Lie algebra

The Euclidean motion Lie algebra $\mathfrak e(3)$ is the real vector space $\mathfrak{so}(3)$ $\oplus\mathbb R^{3}$ with bracket

[(A,v),(B,w)]=(AB-BA,Aw-Bv).

Rotations and translations

Rotations act on the translation vectors by ordinary matrix multiplication. The translations form an abelian ideal; quotienting by them gives the orthogonal Lie algebra. The block-matrix realization $(A,v)\mapsto\begin{pmatrix}A&v\\0&0\end{pmatrix}$ verifies the bracket and Jacobi identity.

The full Euclidean group and its orientation-preserving subgroup have this same tangent Lie algebra; their disconnected components are invisible infinitesimally.