A Neat Rotation Matrix Identity 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?
 A: 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 $h^2$. 
