Consider a unit vector $\hat e = \pmatrix{e_1\cr e_2\cr e_3}$ and an angle $\theta$.
Now construct the following matrix out of of $\hat e$:

\begin{align} M_{\hat e} = \pmatrix{ 0&-e_3&e_2\cr e_3&0&-e_1\cr -e_2&e_1&0} \end{align}

The rotation matrix that rotates by an angle $\theta$ about $\hat e$ is given by

\begin{align} e^{\theta M_{\hat e}} \end{align}

Now, suppose we have two rotations determined by $(\hat e_1, \theta_1)$ and $(\hat e_2, \theta_2)$. Now consider the product of these two rotations

\begin{align} e^{\theta_1 M_{\hat e_1}}e^{\theta_2 M_{\hat e_2}}=e^{\theta_3 M_{\hat e_3}} \end{align}

Can we find $(\hat e_3, \theta_3)$ in terms of $(\hat e_1, \theta_1)$ and $(\hat e_2, \theta_2)$?

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