3.4 Adjoint Representation

$\displaystyle X$	$\displaystyle =\left(\begin{array}{cc} a & -b\\ b & a \end{array}\right)\in$	$\displaystyle \mathrm{SO}(2),a^{2}+b^{2}=1$	(40)
$\displaystyle \mathrm{Adj}_{X}\left(\mathrm{alg}\left(\theta\right)\right)$	$\displaystyle =$	$\displaystyle X\mathrm{\cdot alg}\left(\theta\right)\cdot X^{-1}$	(41)
	$\displaystyle =$	$\displaystyle \left(\begin{array}{cc} 0 & -\theta\\ \theta & 0 \end{array}\right)$	(42)
	$\displaystyle =$	$\displaystyle \mathrm{alg}\left(\theta\right)$	(43)
$\displaystyle \implies\mathrm{Adj}_{X}$	$\displaystyle =$	$\displaystyle \mathbf{I}$	(44)