# Zassenhaus formula for arbitrary (faithful) representation of SU(2)

In Closed form of Baker Campbell Hausdorff theorem with cyclic bracket structure a BCH formula is given which works for an arbitrary faithful representation of SU(2). Now, is there an equivalent relation for the Zassenhaus formula?

More explicitly, can you write $$\exp(it \hat n \cdot \vec\sigma)$$ as a product of exponentials of the Pauli matrices? As argued in the linked post, this would give a relation that works for any faithful representation of SU(2).

• Sounds like you're just asking for an extrinsic Euler angle decomposition of versors. Commented Apr 10, 2022 at 22:57
• I'm not really familiar with the Euler angle formalism; can you maybe give a reference for such a decomposition? Commented Apr 11, 2022 at 13:43
• This Wikipedia page (see also the links therein), and unit norm quaternions are iso to $SU(2)$. That at least all rotations ($SO(3)$) are a chain of three intrinsic rotations is clear from this animation. All details (made bothersome by Euler angles being so practical and thus resources often not being for $SU$) is more work and a full answer might include an algorithm computing angles from a standard rep. Commented Apr 11, 2022 at 20:52