Let $\mathbf{R}_i$ be $N$ rotation matrices that represent a rotation around axes $\mathbf{\omega}_i$ by an angle $|\mathbf{\omega}_i|$. Now say we know that the product of these matrices is unity, i.e.: $$\prod_{i=1}^N \mathbf{R}_i = \mathbf{R}_1\mathbf{R}_2\mathbf{R}_3 \ldots \mathbf{R}_N = \mathbf{I}$$
A paper I'm reading claims, that to a "first approximation" and "omitting the details" the following holds true: $$\mathbf{\omega}_1 +\mathbf{R}_1\mathbf{\omega}_2 +\mathbf{R}_1\mathbf{R}_2\mathbf{\omega}_3 +\mathbf{R}_1\mathbf{R}_2\mathbf{R}_3\mathbf{\omega}_4 + \ldots +\prod_{i=1}^{N-1}\mathbf{R}_i{\omega}_N = \mathbf{0} \tag{**} $$
Now I know that by definition, $\mathbf{R}_i\mathbf{\omega}_i =\mathbf{\omega}_i$, and that one can expand any $\mathbf{R}_i$ in powers of $\mathbf{\omega}_i$: $$\mathbf{R}_i\ = \exp([\mathbf{\omega}_i]_\times) \approx\mathbf{I}+[\mathbf{\omega}_i]_\times+ \frac{1}{2!}[\mathbf{\omega}_i]^2_\times +\ldots$$ But I still can't quite see how one can derive $(**)$. Any ideas?
Suppose all the $|\omega_i|<h$, then we prove inductively that formula (**) holds to order $h^2$.
When $N=2$ you have $R_1 = I+[\omega_1]_\times+O(h^2)$ and $R_2 = I-[\omega_1]_\times+O(h^2)$ so (**) becomes $\omega_1+R_1(-\omega_1) = 0$ to order $h^2$.
Suppose inductively that it holds for $N-1$ factors, and look at a case of $N$ factors $$ R_1R_2\cdots R_{N-2}\big(R_{N-1}R_{N}\big) = I, \tag{1} $$ where I have grouped the last two together; denote the last group as $R'$. By inductive hypothesis we have $$ \omega_1+\cdots+R_1R_2\cdots R_{N-3}R_{N-2}\omega' = 0, \tag{2} $$ where $R' = 1+[\omega']_\times+O(h^2)$. But $$ \omega' = \omega_{N-1}+\omega_N+O(h^2) $$ because $$ R' = R_{N-1}R_N = \big(I+[\omega_{N-1}]_\times+O(h^2)\big) \big(I+[\omega_{N}]_\times+O(h^2)\big) $$ $$ = \big(I+[\omega_{N-1}+\omega_{N}]_\times+O(h^2)\big). $$ Therefore $$ \omega' = R'\omega' = R_{N-1}R_N\omega' = R_{N-1}\big( I+[\omega_N]_\times+O(h^2) \big)\big( \omega_{N-1}+\omega_N+O(h^2) \big) $$ $$ = R_{N-1}(\omega_{N-1}+\omega_N)+O(h^2) = \omega_{N-1}+R_{N-1}\omega_N+O(h^2). $$ Substitute this into (2) to get (**) to order $h^2$.