Refined Question: Decomposition of Rotations in $SU(2)$
In reference to the group compostion law of elements of $SU(2)$, cf. e.g. eq. (13) in Qmechanic's answer here, one thing I do not fully understand is how to decompose a given element of $SU(2)$ (such as a rotation $e^{i \lambda (\mathbf{n} \cdot \boldsymbol{\sigma})}$) into two rotations. Specifically:
Can the respective angles $\lambda_1, \lambda_2$ and axes $\mathbf{n}_1, \mathbf{n}_2$ of the two rotations be chosen arbitrarily as long as they satisfy the composition law for the given element?
Or is there a systematic or unique procedure to determine these parameters, depending on the given element?
