Lie Algebra

The tangent space of $SO(3)$ at the identity is denoted by

$$\mathfrak{so}(3).$$ (A.30)

The elements of $\mathfrak{so}(3)$ are skew-symmetric matrices of the form

$$\boldsymbol{\Omega} = \begin{bmatrix} 0 & -\omega_z & \omega... ...a_z & 0 & -\omega_x\\ -\omega_y & \omega_x & 0 \end{bmatrix}.$$ (A.31)

Introducing the vector

$$\boldsymbol{\omega} = \begin{bmatrix} \omega_x\\ \omega_y\\ \omega_z \end{bmatrix},$$ (A.32)

it is possible to write, in compact form,

$$\boldsymbol{\Omega} = [\boldsymbol{\omega}]_\times ,$$ (A.33)

where $[\cdot]_\times$ represents the matrix associated with the cross product.

It should be noted that the parameterization by infinitesimal rotations introduced in the preceding sections coincides exactly with an element of the Lie algebra $\mathfrak{so}(3)$.