Equivalent Rotation using Baker-Campbell-Hausdorff relation
I) The Baker-Campbell-Hausdorff (BCH) formula for the 3-dimensional rotations can indeed be summed up. Here we will just state the result in the notation of Ref. 1.
Three-dimensional rotations are described by the Lie group $SO(3)$. The corresponding Lie algebra $so(3)$ is
$$ \begin{align} [L_j, L_k] ~=~& i\sum_{\ell=1}^3\epsilon_{jk\ell} L_{\ell}, \cr j,k,\ell~\in~& \{1,2,3\},\cr \epsilon_{123}~=~&1, \cr i^2~=~&-1.\end{align}\tag{1} $$
In adjoint representation, the three Lie algebra generators $iL_{\ell}\in{\rm Mat}_{3\times 3}(\mathbb{R})$, $\ell\in\{1,2,3\}$, are $3\times 3$ real antisymmetric matrices,
$$ \begin{align}i(L_j)_{k\ell} ~=~& \epsilon_{jk\ell} ,\cr j,k,\ell~\in~& \{1,2,3\}.\end{align}\tag{2} $$
II) A rotation matrix
$$ R(\vec{\alpha})~\in~ SO(3)~ \subseteq ~{\rm Mat}_{3\times 3}(\mathbb{R})\tag{3}$$
can be specified by a rotation axis and an rotation angle. Here we will use a 3-vector
$$ \vec{\alpha}~=~\alpha \vec{n}_{\alpha}~\in~ \mathbb{R}^3,\tag{4}$$
where $\vec{n}_{\alpha}\in\mathbb{R}^3$ is a unit vector parallel to the rotation axis, $|\vec{n}_{\alpha}|=1$ ; and $\alpha\in \mathbb{R}$ (without an arrow on top) is the angle of rotation.
Define for later convenience
$$ c_{\alpha} ~:=~\cos(\alpha) ~\in~ \mathbb{R},\tag{5} $$ $$\vec{s}_{\alpha}~:=~\sin(\alpha)\vec{n}_{\alpha} ~\in~ \mathbb{R}^3,\tag{6} $$ $$ \vec{t}_{\alpha}~:=~\tan(\alpha)\vec{n}_{\alpha} ~\in~ \mathbb{R}^3.\tag{7} $$
The formula for the rotation matrix in terms of $\vec{\alpha}$ reads
$$\begin{align} R(\vec{\alpha}) ~=~&e^{i \vec{\alpha}\cdot \vec{L}}\cr ~=~& {\bf 1}_{3\times 3} - (1-c_{\alpha}) (\vec{n}_{\alpha}\cdot \vec{L})^2 + i\vec{s}_{\alpha}\cdot \vec{L}.\end{align} \tag{8}$$
III) The composition of two rotations is again a rotation
$$ R(\vec{\gamma})~=~R(\vec{\alpha})R(\vec{\beta}).\tag{9}$$
The "addition formula" for the corresponding $3$-vectors reads
$$ \vec{t}_{\gamma} ~=~\frac{\vec{t}_{\alpha}+\vec{t}_{\beta}-\vec{t}_{\alpha}\times\vec{t}_{\beta} }{1-\vec{t}_{\alpha}\cdot \vec{t}_{\beta}}.\tag{10} $$
IV) The derivation of eq. (10) simplifies if one uses the fact that $SU(2)\cong U(1,\mathbb{H})$ is the double cover of $SO(3)$. An $SU(2)$-matrix
$$ X(\vec{\alpha})~\in~ SU(2)~ \subseteq ~{\rm Mat}_{2\times 2}(\mathbb{C})\tag{11}$$
can be written in terms of the Pauli matrices as
$$\begin{align} X(\vec{\alpha}) ~=~&e^{i \vec{\alpha}\cdot \vec{\sigma}/2}\cr ~=~& c_{\alpha}{\bf 1}_{2\times 2} + i\vec{s}_{\alpha}\cdot \vec{\sigma}.\end{align} \tag{12}$$
The composition of two $SU(2)$-matrix is given by the same BCH formula
$$ X(\vec{\gamma})~=~X(\vec{\alpha})X(\vec{\beta}).\tag{13}$$
References:
G 't Hooft, Introduction to Lie Groups in Physics, lecture notes, chapter 3. The pdf file is available here.
S. Weigert, J. Phys. A30 (1997) 8739, arXiv:quant-ph/9710024.
K. Engø, On the BCH-formula in so(3), Bit Num. Math. 41 (2001) 629. (Hat tip: WetSavannaAnimal aka Rod Vance.)
Wikipedia.