Prove $(1/(a_{23}+a_{32}),1/(a_{13}+a_{31}),1/(a_{12}+a_{21}))$ is an eigenvector of $A \in SO(3)$ to the eigenvalues one 
Prove $$\begin{bmatrix}\frac1{a_{23}+a_{32}}\\\frac1{a_{13}+a_{31}}\\\frac1{a_{12}+a_{21}}\end{bmatrix}$$ is an eigenvector of $A \in SO(3)$ to the eigenvalue one if the eigenvector even exists in the first place.

The only approach that I've found is the use the property $\det(A)=1$ and $A A^{T}=I$ figure out equations systems for $a_{ij}$ and use these in order to show algebraically $Av=v$. But I know from practical experience that this will take a long time and gets also really ugly.
 A: If $A$ is a rotation matrix, then the eigenvector corresponding to $\lambda = 1$ is along the axis of rotation.
Now, on the other hand, a rotation matrix can be decomposed using the Rodrigues' rotation matrix formula as follows
$A = {a a}^T + (I - {aa}^T ) \cos \theta + S_a \sin \theta $
where $a= [a_x, a_y, a_z]^T $ is the unit vector along the axis of rotation, $\theta $ is the angle of rotation, and the skew symmetric matrix $S_a$ is given by
$S_a = \begin{bmatrix} 0 && - a_z && a_y \\ a_z && 0 && - a_x \\ - a_y && a_x && 0 \end{bmatrix} $
From this, and noting the pattern on the off-diagonal elements of the rotation matrix $A$ we find that
$ A_{12} + A_{21} = 2 (a_x a_y) (1 - \cos \theta)$
$ A_{13} + A_{31} = 2 (a_x a_z) (1 - \cos \theta)$
$ A_{23} + A_{32} = 2 (a_y a_z) ( 1 - \cos \theta)$
So, assuming $ \cos \theta \ne 1 $ i.e. $\theta \ne 0 $, then we can write the above as
$ a_z(A_{12} + A_{21}) = 2 (a_x a_y a_z) (1 - \cos \theta)$
$ a_y( A_{13} + A_{31} ) = 2 (a_x a_y a_z) (1 - \cos \theta)$
$ a_x(A_{23} + A_{32} ) = 2 (a_x a_y a_z) ( 1 - \cos \theta)$
Define $K = 2 (a_x a_y a_z) (1 - \cos \theta) $
Then
$a_x = \dfrac{K}{A_{23} + A_{32}} $
$a_y = \dfrac{K}{A_{13} + A_{31} } $
$a_z = \dfrac{K}{A_{12} + A_{21} } $
Therefore, the vector
$\left( \dfrac{1}{A_{23} + A_{32}} ,
 \dfrac{1}{A_{13} + A_{31} } ,\dfrac{1}{A_{12} + A_{21} } \right) = \dfrac{1}{K} (a_x, a_y, a_z) $
is along the axis unit vector $(a_x, a_y, a_z)$ and is therefore an eigenvector of the rotation matrix $A$ corresponding to $\lambda = 1 $.
