8.4 Adjoint Representation

The adjoint representation of $\mathrm{SO}(3)$ is actually identical to the rotation matrix representation due to properties of the cross product:

$\displaystyle \mathbf{R}$	$\displaystyle \in$	$\displaystyle \mathrm{SO}(3)$	(104)
$\displaystyle \mathbf{a},\mathbf{b}$	$\displaystyle \in$	$\displaystyle \mathbb{R}^{3}$	(105)
$\displaystyle \left(\mathbf{R}\cdot\mathbf{a}_{\times}\cdot\mathbf{R}^{T}\right)\cdot\mathbf{b}$	$\displaystyle =$	$\displaystyle \mathbf{R}\cdot\left(\mathbf{a}\times\mathbf{R}^{T}\cdot\mathbf{b}\right)$	(106)
	$\displaystyle =$	$\displaystyle \left(\mathbf{R}\cdot\mathbf{a}\right)\times\mathbf{b}$	(107)
	$\displaystyle =$	$\displaystyle \left(\mathbf{R}\cdot\mathbf{a}\right)_{\times}\cdot\mathbf{b}$	(108)
$\displaystyle \implies\mathbf{R}\cdot\mathbf{a}_{\times}\cdot\mathbf{R}^{T}$	$\displaystyle =$	$\displaystyle \left(\mathbf{R}\cdot\mathbf{a}\right)_{\times}$	(109)
$\displaystyle \implies\mathrm{Adj}_{\mathbf{R}}$	$\displaystyle =$	$\displaystyle \mathbf{R}$	(110)