With curiousity, I'm trying to prove whether multiplication of all rotation matrixes in $\mathrm{SO}(3)$ is identity irrelevant of multiplication order. As each rotation matrix in $\mathrm{SO}(3)$ space has an inverse rotation matrix in that space, if I multiply them ordered as $R_1*R_1'*R_2*R_2'*....R_n*R_n' = I $ would result in identity. However, if the multiplication order is random, is it possible that the result will be identity?
I tried to prove this in 2D rotation matrices. But because there is only one angle, it seems rotation operation has commutation property in this case. However in $\mathrm{SO}(3)$, rotation using three angles does not have commutation.
In $\mathrm{SO}(3)$, I tried taking logarithm of both sides and writing the multiplication as summation. Then I tried to convert it to triple integrals (for each infinitesimal angle) but I could not go on further from there.